Count numbers built from digit cards

4.NBT.A.2 · take · grade 4

Archetype: Build the Largest or Smallest Value from Digit Cards · step in a 7-type progression

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.

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

Show solution

Understand

Using the four digit cards 8, 0, 5, and 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, and 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.

Plan

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

Whether a number is greater than 5000 is decided almost entirely by the thousands 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.

Execute

#7 Identify Subproblems 4.NBT.A.2

A four-digit number is greater than 5000 when its thousands digit is 5 or more. From the cards 8, 0, 5, 3, only 5 and 8 qualify. (3 gives 3xxx and 0 cannot lead a four-digit number.)

\text{thousands digit} \geq 5 \Rightarrow 5 \text{ or } 8

Comparing multi-digit numbers starts from the highest place, so the thousands digit alone tells you if you have cleared 5000.

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

Put 8 first. The remaining cards 0, 5, 3 fill the hundreds, tens, and ones places in any order. There are 3 x 2 x 1 = 6 orderings, and every one is at least 8035, which is greater than 5000.

3 \times 2 \times 1 = 6

Three different cards into three slots gives 6 different orderings, and starting with 8 guarantees the number passes 5000.

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

Put 5 first. The remaining cards 0, 8, 3 fill the other three places in any order, giving 6 orderings. The smallest is 5038, which is still greater than 5000, so all 6 count.

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

Even the smallest number that starts with 5 here, 5038, is past 5000, so none of these 6 are lost.

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

Combine the two thousands-digit cases: 6 numbers from the 8 group plus 6 numbers from the 5 group.

6 + 6 = 12

The two starting digits give separate, non-overlapping lists, so we simply add the counts.

Answer: 12

Review

There are 18 four-digit numbers in all using these cards (3 choices for a nonzero leading digit times 6 arrangements). Only those starting with 5 or 8 beat 5000, which is exactly 12 of them, a believable fraction.

You could write out the full systematic list of all four-digit numbers and cross off each one that is 5000 or less, then count what remains.

Standards · min 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.

💡 Look at the biggest place first: only cards 5 and 8 can lead a number past 5000, then just count the orderings!