Practice · Count numbers built from digit cards · Sensim Math

Variant 1 easy

Using each number card exactly once, you want to make four-digit numbers that are greater than $5000$ . Find how many such numbers can be made in all.

$\boxed{7}\ \boxed{0}\ \boxed{4}\ \boxed{2}$

Your answer

Show solution

1 · Understandwhat's really being asked

Using the four digit cards 7, 0, 4, 2 each exactly once, build four-digit numbers and count how many of them are greater than 5000.

Givens

The four digit cards are 7, 0, 4, 2.
Each card must be used exactly once to form a four-digit number.
We only count numbers that are greater than 5000.

Unknowns

How many four-digit numbers greater than 5000 can be made.

Constraints

Every number is four digits, so the thousands place cannot be 0.
The number must be strictly greater than 5000.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems

Whether a number beats the threshold is decided almost entirely by the leading digit, so I split the count by which card sits in the thousands place (a subproblem), then systematically list the arrangements of the remaining three cards.

3 · Execute3 carry out the plan

1Decide which card can lead the number

#7 Identify Subproblems 4.NBT.A.2

A four-digit number's size is decided first by its thousands (leading) digit, and 0 cannot lead a four-digit number. Comparing each possible leading card against 5000, the cards that can produce a number greater than 5000 are 7.

\text{leading card} \Rightarrow 7

Comparing multi-digit numbers starts from the highest place, so the leading digit tells you most of the story.

2Count arrangements led by 7

#2 Make a Systematic List 4.NBT.A.2

Put 7 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 7024, which is still greater than 5000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 7024 > 5000

Three different cards into three slots give 6 different orderings.

3Add the cases

#2 Make a Systematic List 4.NBT.A.2

Combine the separate, non-overlapping leading-digit cases.

6 = 6

Each leading digit gives its own list, so we simply add the counts.

Answer: 6

4 · Reviewdoes it hold up?

There are 18 four-digit numbers in all using these cards. Counting only those greater than 5000 gives 6, a believable fraction of the total.

Another way: Write out the full systematic list of all 18 four-digit numbers and cross off each one that is 5000 or less, then count what remains.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing each four-digit number against 5000 by reading the thousands place first.

💡Takeaway. Look at the biggest place first: only some cards can lead a number past the threshold, then just count the orderings!

Variant 2 easy

Using each number card exactly once, you want to make four-digit numbers that are greater than $6000$ . Find how many such numbers can be made in all.

$\boxed{5}\ \boxed{2}\ \boxed{8}\ \boxed{0}$

Your answer

Show solution

1 · Understandwhat's really being asked

Using the four digit cards 5, 2, 8, 0 each exactly once, build four-digit numbers and count how many of them are greater than 6000.

Givens

The four digit cards are 5, 2, 8, 0.
Each card must be used exactly once to form a four-digit number.
We only count numbers that are greater than 6000.

Unknowns

How many four-digit numbers greater than 6000 can be made.

Constraints

Every number is four digits, so the thousands place cannot be 0.
The number must be strictly greater than 6000.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems

Whether a number beats the threshold is decided almost entirely by the leading digit, so I split the count by which card sits in the thousands place (a subproblem), then systematically list the arrangements of the remaining three cards.

3 · Execute3 carry out the plan

1Decide which card can lead the number

#7 Identify Subproblems 4.NBT.A.2

A four-digit number's size is decided first by its thousands (leading) digit, and 0 cannot lead a four-digit number. Comparing each possible leading card against 6000, the cards that can produce a number greater than 6000 are 8.

\text{leading card} \Rightarrow 8

Comparing multi-digit numbers starts from the highest place, so the leading digit tells you most of the story.

2Count arrangements led by 8

#2 Make a Systematic List 4.NBT.A.2

Put 8 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 8025, which is still greater than 6000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 8025 > 6000

Three different cards into three slots give 6 different orderings.

3Add the cases

#2 Make a Systematic List 4.NBT.A.2

Combine the separate, non-overlapping leading-digit cases.

6 = 6

Each leading digit gives its own list, so we simply add the counts.

Answer: 6

4 · Reviewdoes it hold up?

There are 18 four-digit numbers in all using these cards. Counting only those greater than 6000 gives 6, a believable fraction of the total.

Another way: Write out the full systematic list of all 18 four-digit numbers and cross off each one that is 6000 or less, then count what remains.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing each four-digit number against 6000 by reading the thousands place first.

💡Takeaway. Look at the biggest place first: only some cards can lead a number past the threshold, then just count the orderings!

Variant 3 easy

Using each number card exactly once, you want to make four-digit numbers that are greater than $7000$ . Find how many such numbers can be made in all.

$\boxed{6}\ \boxed{0}\ \boxed{2}\ \boxed{9}$

Your answer

Show solution

1 · Understandwhat's really being asked

Using the four digit cards 6, 0, 2, 9 each exactly once, build four-digit numbers and count how many of them are greater than 7000.

Givens

The four digit cards are 6, 0, 2, 9.
Each card must be used exactly once to form a four-digit number.
We only count numbers that are greater than 7000.

Unknowns

How many four-digit numbers greater than 7000 can be made.

Constraints

Every number is four digits, so the thousands place cannot be 0.
The number must be strictly greater than 7000.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems

Whether a number beats the threshold is decided almost entirely by the leading digit, so I split the count by which card sits in the thousands place (a subproblem), then systematically list the arrangements of the remaining three cards.

3 · Execute3 carry out the plan

1Decide which card can lead the number

#7 Identify Subproblems 4.NBT.A.2

A four-digit number's size is decided first by its thousands (leading) digit, and 0 cannot lead a four-digit number. Comparing each possible leading card against 7000, the cards that can produce a number greater than 7000 are 9.

\text{leading card} \Rightarrow 9

Comparing multi-digit numbers starts from the highest place, so the leading digit tells you most of the story.

2Count arrangements led by 9

#2 Make a Systematic List 4.NBT.A.2

Put 9 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 9026, which is still greater than 7000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 9026 > 7000

Three different cards into three slots give 6 different orderings.

3Add the cases

#2 Make a Systematic List 4.NBT.A.2

Combine the separate, non-overlapping leading-digit cases.

6 = 6

Each leading digit gives its own list, so we simply add the counts.

Answer: 6

4 · Reviewdoes it hold up?

There are 18 four-digit numbers in all using these cards. Counting only those greater than 7000 gives 6, a believable fraction of the total.

Another way: Write out the full systematic list of all 18 four-digit numbers and cross off each one that is 7000 or less, then count what remains.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing each four-digit number against 7000 by reading the thousands place first.

💡Takeaway. Look at the biggest place first: only some cards can lead a number past the threshold, then just count the orderings!

Variant 4 medium

Using each number card exactly once, you want to make four-digit numbers that are greater than $8000$ . Find how many such numbers can be made in all.

$\boxed{9}\ \boxed{0}\ \boxed{4}\ \boxed{7}$

Your answer

Show solution

1 · Understandwhat's really being asked

Using the four digit cards 9, 0, 4, 7 each exactly once, build four-digit numbers and count how many of them are greater than 8000.

Givens

The four digit cards are 9, 0, 4, 7.
Each card must be used exactly once to form a four-digit number.
We only count numbers that are greater than 8000.

Unknowns

How many four-digit numbers greater than 8000 can be made.

Constraints

Every number is four digits, so the thousands place cannot be 0.
The number must be strictly greater than 8000.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems

Whether a number beats the threshold is decided almost entirely by the leading digit, so I split the count by which card sits in the thousands place (a subproblem), then systematically list the arrangements of the remaining three cards.

3 · Execute3 carry out the plan

1Decide which card can lead the number

#7 Identify Subproblems 4.NBT.A.2

A four-digit number's size is decided first by its thousands (leading) digit, and 0 cannot lead a four-digit number. Comparing each possible leading card against 8000, the cards that can produce a number greater than 8000 are 9.

\text{leading card} \Rightarrow 9

Comparing multi-digit numbers starts from the highest place, so the leading digit tells you most of the story.

2Count arrangements led by 9

#2 Make a Systematic List 4.NBT.A.2

Put 9 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 9047, which is still greater than 8000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 9047 > 8000

Three different cards into three slots give 6 different orderings.

3Add the cases

#2 Make a Systematic List 4.NBT.A.2

Combine the separate, non-overlapping leading-digit cases.

6 = 6

Each leading digit gives its own list, so we simply add the counts.

Answer: 6

4 · Reviewdoes it hold up?

There are 18 four-digit numbers in all using these cards. Counting only those greater than 8000 gives 6, a believable fraction of the total.

Another way: Write out the full systematic list of all 18 four-digit numbers and cross off each one that is 8000 or less, then count what remains.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing each four-digit number against 8000 by reading the thousands place first.

💡Takeaway. Look at the biggest place first: only some cards can lead a number past the threshold, then just count the orderings!

Variant 5 medium

Using each number card exactly once, you want to make four-digit numbers that are greater than $3000$ . Find how many such numbers can be made in all.

$\boxed{4}\ \boxed{7}\ \boxed{0}\ \boxed{1}$

Your answer

Show solution

1 · Understandwhat's really being asked

Using the four digit cards 4, 7, 0, 1 each exactly once, build four-digit numbers and count how many of them are greater than 3000.

Givens

The four digit cards are 4, 7, 0, 1.
Each card must be used exactly once to form a four-digit number.
We only count numbers that are greater than 3000.

Unknowns

How many four-digit numbers greater than 3000 can be made.

Constraints

Every number is four digits, so the thousands place cannot be 0.
The number must be strictly greater than 3000.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems

Whether a number beats the threshold is decided almost entirely by the leading digit, so I split the count by which card sits in the thousands place (a subproblem), then systematically list the arrangements of the remaining three cards.

3 · Execute4 carry out the plan

1Decide which card can lead the number

#7 Identify Subproblems 4.NBT.A.2

A four-digit number's size is decided first by its thousands (leading) digit, and 0 cannot lead a four-digit number. Comparing each possible leading card against 3000, the cards that can produce a number greater than 3000 are 4, 7.

\text{leading card} \Rightarrow 4, 7

Comparing multi-digit numbers starts from the highest place, so the leading digit tells you most of the story.

2Count arrangements led by 4

#2 Make a Systematic List 4.NBT.A.2

Put 4 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 4017, which is still greater than 3000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 4017 > 3000

Three different cards into three slots give 6 different orderings.

3Count arrangements led by 7

#2 Make a Systematic List 4.NBT.A.2

Put 7 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 7014, which is still greater than 3000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 7014 > 3000

Three different cards into three slots give 6 different orderings.

4Add the cases

#2 Make a Systematic List 4.NBT.A.2

Combine the separate, non-overlapping leading-digit cases.

6 + 6 = 12

Each leading digit gives its own list, so we simply add the counts.

Answer: 12

4 · Reviewdoes it hold up?

There are 18 four-digit numbers in all using these cards. Counting only those greater than 3000 gives 12, a believable fraction of the total.

Another way: Write out the full systematic list of all 18 four-digit numbers and cross off each one that is 3000 or less, then count what remains.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing each four-digit number against 3000 by reading the thousands place first.

💡Takeaway. Look at the biggest place first: only some cards can lead a number past the threshold, then just count the orderings!

Variant 6 medium

Using each number card exactly once, you want to make four-digit numbers that are greater than $4000$ . Find how many such numbers can be made in all.

$\boxed{2}\ \boxed{7}\ \boxed{5}\ \boxed{0}$

Your answer

Show solution

1 · Understandwhat's really being asked

Using the four digit cards 2, 7, 5, 0 each exactly once, build four-digit numbers and count how many of them are greater than 4000.

Givens

The four digit cards are 2, 7, 5, 0.
Each card must be used exactly once to form a four-digit number.
We only count numbers that are greater than 4000.

Unknowns

How many four-digit numbers greater than 4000 can be made.

Constraints

Every number is four digits, so the thousands place cannot be 0.
The number must be strictly greater than 4000.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems

Whether a number beats the threshold is decided almost entirely by the leading digit, so I split the count by which card sits in the thousands place (a subproblem), then systematically list the arrangements of the remaining three cards.

3 · Execute4 carry out the plan

1Decide which card can lead the number

#7 Identify Subproblems 4.NBT.A.2

A four-digit number's size is decided first by its thousands (leading) digit, and 0 cannot lead a four-digit number. Comparing each possible leading card against 4000, the cards that can produce a number greater than 4000 are 5, 7.

\text{leading card} \Rightarrow 5, 7

Comparing multi-digit numbers starts from the highest place, so the leading digit tells you most of the story.

2Count arrangements led by 5

#2 Make a Systematic List 4.NBT.A.2

Put 5 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 5027, which is still greater than 4000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 5027 > 4000

Three different cards into three slots give 6 different orderings.

3Count arrangements led by 7

#2 Make a Systematic List 4.NBT.A.2

Put 7 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 7025, which is still greater than 4000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 7025 > 4000

Three different cards into three slots give 6 different orderings.

4Add the cases

#2 Make a Systematic List 4.NBT.A.2

Combine the separate, non-overlapping leading-digit cases.

6 + 6 = 12

Each leading digit gives its own list, so we simply add the counts.

Answer: 12

4 · Reviewdoes it hold up?

There are 18 four-digit numbers in all using these cards. Counting only those greater than 4000 gives 12, a believable fraction of the total.

Another way: Write out the full systematic list of all 18 four-digit numbers and cross off each one that is 4000 or less, then count what remains.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing each four-digit number against 4000 by reading the thousands place first.

💡Takeaway. Look at the biggest place first: only some cards can lead a number past the threshold, then just count the orderings!

Variant 7 medium

Using each number card exactly once, you want to make four-digit numbers that are greater than $4000$ . Find how many such numbers can be made in all.

$\boxed{3}\ \boxed{8}\ \boxed{1}\ \boxed{5}$

Your answer

Show solution

1 · Understandwhat's really being asked

Using the four digit cards 3, 8, 1, 5 each exactly once, build four-digit numbers and count how many of them are greater than 4000.

Givens

The four digit cards are 3, 8, 1, 5.
Each card must be used exactly once to form a four-digit number.
We only count numbers that are greater than 4000.

Unknowns

How many four-digit numbers greater than 4000 can be made.

Constraints

Every number is four digits, so the thousands place cannot be 0.
The number must be strictly greater than 4000.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems

Whether a number beats the threshold is decided almost entirely by the leading digit, so I split the count by which card sits in the thousands place (a subproblem), then systematically list the arrangements of the remaining three cards.

3 · Execute4 carry out the plan

1Decide which card can lead the number

#7 Identify Subproblems 4.NBT.A.2

A four-digit number's size is decided first by its thousands (leading) digit, and 0 cannot lead a four-digit number. Comparing each possible leading card against 4000, the cards that can produce a number greater than 4000 are 5, 8.

\text{leading card} \Rightarrow 5, 8

Comparing multi-digit numbers starts from the highest place, so the leading digit tells you most of the story.

2Count arrangements led by 5

#2 Make a Systematic List 4.NBT.A.2

Put 5 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 5138, which is still greater than 4000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 5138 > 4000

Three different cards into three slots give 6 different orderings.

3Count arrangements led by 8

#2 Make a Systematic List 4.NBT.A.2

Put 8 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 8135, which is still greater than 4000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 8135 > 4000

Three different cards into three slots give 6 different orderings.

4Add the cases

#2 Make a Systematic List 4.NBT.A.2

Combine the separate, non-overlapping leading-digit cases.

6 + 6 = 12

Each leading digit gives its own list, so we simply add the counts.

Answer: 12

4 · Reviewdoes it hold up?

There are 24 four-digit numbers in all using these cards. Counting only those greater than 4000 gives 12, a believable fraction of the total.

Another way: Write out the full systematic list of all 24 four-digit numbers and cross off each one that is 4000 or less, then count what remains.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing each four-digit number against 4000 by reading the thousands place first.

💡Takeaway. Look at the biggest place first: only some cards can lead a number past the threshold, then just count the orderings!

Variant 8 medium

Using each number card exactly once, you want to make four-digit numbers that are greater than $5000$ . Find how many such numbers can be made in all.

$\boxed{8}\ \boxed{0}\ \boxed{5}\ \boxed{3}$

Your answer

Show solution

1 · Understandwhat's really being asked

Using the four digit cards 8, 0, 5, 3 each exactly once, build four-digit numbers and count how many of them are greater than 5000.

Givens

The four digit cards are 8, 0, 5, 3.
Each card must be used exactly once to form a four-digit number.
We only count numbers that are greater than 5000.

Unknowns

How many four-digit numbers greater than 5000 can be made.

Constraints

Every number is four digits, so the thousands place cannot be 0.
The number must be strictly greater than 5000.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems

Whether a number beats the threshold is decided almost entirely by the leading digit, so I split the count by which card sits in the thousands place (a subproblem), then systematically list the arrangements of the remaining three cards.

3 · Execute4 carry out the plan

1Decide which card can lead the number

#7 Identify Subproblems 4.NBT.A.2

A four-digit number's size is decided first by its thousands (leading) digit, and 0 cannot lead a four-digit number. Comparing each possible leading card against 5000, the cards that can produce a number greater than 5000 are 5, 8.

\text{leading card} \Rightarrow 5, 8

Comparing multi-digit numbers starts from the highest place, so the leading digit tells you most of the story.

2Count arrangements led by 5

#2 Make a Systematic List 4.NBT.A.2

Put 5 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 5038, which is still greater than 5000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 5038 > 5000

Three different cards into three slots give 6 different orderings.

3Count arrangements led by 8

#2 Make a Systematic List 4.NBT.A.2

Put 8 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 8035, which is still greater than 5000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 8035 > 5000

Three different cards into three slots give 6 different orderings.

4Add the cases

#2 Make a Systematic List 4.NBT.A.2

Combine the separate, non-overlapping leading-digit cases.

6 + 6 = 12

Each leading digit gives its own list, so we simply add the counts.

Answer: 12

4 · Reviewdoes it hold up?

There are 18 four-digit numbers in all using these cards. Counting only those greater than 5000 gives 12, a believable fraction of the total.

Another way: Write out the full systematic list of all 18 four-digit numbers and cross off each one that is 5000 or less, then count what remains.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing each four-digit number against 5000 by reading the thousands place first.

💡Takeaway. Look at the biggest place first: only some cards can lead a number past the threshold, then just count the orderings!

Variant 9 hard

Using each number card exactly once, you want to make four-digit numbers that are greater than $5000$ . Find how many such numbers can be made in all.

$\boxed{1}\ \boxed{9}\ \boxed{4}\ \boxed{6}$

Your answer

Show solution

1 · Understandwhat's really being asked

Using the four digit cards 1, 9, 4, 6 each exactly once, build four-digit numbers and count how many of them are greater than 5000.

Givens

The four digit cards are 1, 9, 4, 6.
Each card must be used exactly once to form a four-digit number.
We only count numbers that are greater than 5000.

Unknowns

How many four-digit numbers greater than 5000 can be made.

Constraints

Every number is four digits, so the thousands place cannot be 0.
The number must be strictly greater than 5000.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems

Whether a number beats the threshold is decided almost entirely by the leading digit, so I split the count by which card sits in the thousands place (a subproblem), then systematically list the arrangements of the remaining three cards.

3 · Execute4 carry out the plan

1Decide which card can lead the number

#7 Identify Subproblems 4.NBT.A.2

A four-digit number's size is decided first by its thousands (leading) digit, and 0 cannot lead a four-digit number. Comparing each possible leading card against 5000, the cards that can produce a number greater than 5000 are 6, 9.

\text{leading card} \Rightarrow 6, 9

Comparing multi-digit numbers starts from the highest place, so the leading digit tells you most of the story.

2Count arrangements led by 6

#2 Make a Systematic List 4.NBT.A.2

Put 6 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 6149, which is still greater than 5000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 6149 > 5000

Three different cards into three slots give 6 different orderings.

3Count arrangements led by 9

#2 Make a Systematic List 4.NBT.A.2

Put 9 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 9146, which is still greater than 5000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 9146 > 5000

Three different cards into three slots give 6 different orderings.

4Add the cases

#2 Make a Systematic List 4.NBT.A.2

Combine the separate, non-overlapping leading-digit cases.

6 + 6 = 12

Each leading digit gives its own list, so we simply add the counts.

Answer: 12

4 · Reviewdoes it hold up?

There are 24 four-digit numbers in all using these cards. Counting only those greater than 5000 gives 12, a believable fraction of the total.

Another way: Write out the full systematic list of all 24 four-digit numbers and cross off each one that is 5000 or less, then count what remains.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing each four-digit number against 5000 by reading the thousands place first.

💡Takeaway. Look at the biggest place first: only some cards can lead a number past the threshold, then just count the orderings!

Variant 10 hard

Using each number card exactly once, you want to make four-digit numbers that are greater than $6000$ . Find how many such numbers can be made in all.

$\boxed{9}\ \boxed{1}\ \boxed{6}\ \boxed{0}$

Your answer

Show solution

1 · Understandwhat's really being asked

Using the four digit cards 9, 1, 6, 0 each exactly once, build four-digit numbers and count how many of them are greater than 6000.

Givens

The four digit cards are 9, 1, 6, 0.
Each card must be used exactly once to form a four-digit number.
We only count numbers that are greater than 6000.

Unknowns

How many four-digit numbers greater than 6000 can be made.

Constraints

Every number is four digits, so the thousands place cannot be 0.
The number must be strictly greater than 6000.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems

Whether a number beats the threshold is decided almost entirely by the leading digit, so I split the count by which card sits in the thousands place (a subproblem), then systematically list the arrangements of the remaining three cards.

3 · Execute4 carry out the plan

1Decide which card can lead the number

#7 Identify Subproblems 4.NBT.A.2

A four-digit number's size is decided first by its thousands (leading) digit, and 0 cannot lead a four-digit number. Comparing each possible leading card against 6000, the cards that can produce a number greater than 6000 are 6, 9.

\text{leading card} \Rightarrow 6, 9

Comparing multi-digit numbers starts from the highest place, so the leading digit tells you most of the story.

2Count arrangements led by 6

#2 Make a Systematic List 4.NBT.A.2

Put 6 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 6019, which is still greater than 6000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 6019 > 6000

Three different cards into three slots give 6 different orderings.

3Count arrangements led by 9

#2 Make a Systematic List 4.NBT.A.2

Put 9 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 9016, which is still greater than 6000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 9016 > 6000

Three different cards into three slots give 6 different orderings.

4Add the cases

#2 Make a Systematic List 4.NBT.A.2

Combine the separate, non-overlapping leading-digit cases.

6 + 6 = 12

Each leading digit gives its own list, so we simply add the counts.

Answer: 12

4 · Reviewdoes it hold up?

There are 18 four-digit numbers in all using these cards. Counting only those greater than 6000 gives 12, a believable fraction of the total.

Another way: Write out the full systematic list of all 18 four-digit numbers and cross off each one that is 6000 or less, then count what remains.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing each four-digit number against 6000 by reading the thousands place first.

💡Takeaway. Look at the biggest place first: only some cards can lead a number past the threshold, then just count the orderings!

Variant 11 hard

Using each number card exactly once, you want to make four-digit numbers that are greater than $6000$ . Find how many such numbers can be made in all.

$\boxed{8}\ \boxed{3}\ \boxed{1}\ \boxed{6}$

Your answer

Show solution

1 · Understandwhat's really being asked

Using the four digit cards 8, 3, 1, 6 each exactly once, build four-digit numbers and count how many of them are greater than 6000.

Givens

The four digit cards are 8, 3, 1, 6.
Each card must be used exactly once to form a four-digit number.
We only count numbers that are greater than 6000.

Unknowns

How many four-digit numbers greater than 6000 can be made.

Constraints

Every number is four digits, so the thousands place cannot be 0.
The number must be strictly greater than 6000.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems

Whether a number beats the threshold is decided almost entirely by the leading digit, so I split the count by which card sits in the thousands place (a subproblem), then systematically list the arrangements of the remaining three cards.

3 · Execute4 carry out the plan

1Decide which card can lead the number

#7 Identify Subproblems 4.NBT.A.2

A four-digit number's size is decided first by its thousands (leading) digit, and 0 cannot lead a four-digit number. Comparing each possible leading card against 6000, the cards that can produce a number greater than 6000 are 6, 8.

\text{leading card} \Rightarrow 6, 8

Comparing multi-digit numbers starts from the highest place, so the leading digit tells you most of the story.

2Count arrangements led by 6

#2 Make a Systematic List 4.NBT.A.2

Put 6 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 6138, which is still greater than 6000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 6138 > 6000

Three different cards into three slots give 6 different orderings.

3Count arrangements led by 8

#2 Make a Systematic List 4.NBT.A.2

Put 8 first. The remaining three cards fill the other places in any order, giving 3 x 2 x 1 = 6 orderings. The smallest of them is 8136, which is still greater than 6000, so all 6 count.

3 \times 2 \times 1 = 6,\quad 8136 > 6000

Three different cards into three slots give 6 different orderings.

4Add the cases

#2 Make a Systematic List 4.NBT.A.2

Combine the separate, non-overlapping leading-digit cases.

6 + 6 = 12

Each leading digit gives its own list, so we simply add the counts.

Answer: 12

4 · Reviewdoes it hold up?

There are 24 four-digit numbers in all using these cards. Counting only those greater than 6000 gives 12, a believable fraction of the total.

Another way: Write out the full systematic list of all 24 four-digit numbers and cross off each one that is 6000 or less, then count what remains.

Standardsmin grade 4

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Comparing each four-digit number against 6000 by reading the thousands place first.

💡Takeaway. Look at the biggest place first: only some cards can lead a number past the threshold, then just count the orderings!

Count numbers built from digit cards · 11 practice problems

Generated variants — 11

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