Problem
Count numbers built from digit cards
Using cards 8·0·5·3 each exactly once, how many four-digit numbers are greater than 5000?
Base-ten numbers
Your answer
numbers
Watch out It is a four-digit number, so the thousands place can't be 0
How to solve
1STEP 1
Start with the thousands place
To beat 5000 the thousands digit must be 5 or more — only 5 and 8 work. (0 can't lead)
5 □ □ □ · 8 □ □ □
The thousands place is worth 1000, but the three lower places together reach only 999, so the thousands digit alone decides whether you clear 5000.
Identify SubproblemsFour Digit Numbers4.NBT.A.2A four-digit number built from the cards 8, 0, 5, and 3 is greater than 5000 exactly when the card in the thousands place is 5 or 8.
Why?
Whether a four-digit number clears 5000 is settled by its thousands digit alone, because that place is worth a whole thousand.
Why?
Moving the thousands digit up by one adds a full 1000, while the hundreds, tens, and ones places together can only ever reach 999, so they can never make up for a smaller thousands digit.
Why?
Each place holds ten of the place beneath it, so one thousand is bigger than any amount the three lower places can pile up.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
Of the four cards, only 5 and 8 are large enough in the thousands place to pass 5000, while 3 keeps the number down in the 3000s and 0 cannot lead a four-digit number at all.
Why?
Because the thousands digit decides the size, a 3 there gives a number below 4000, and a 0 there means there is no thousands at all, so the leading card has to be 5 or 8.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
2STEP 2
List the rest, none missed
The three remaining cards arrange 6 ways — both cases stay above 5000.
3 × 2 × 1 = 6
Three choices for the first slot, two for the next, one left for the last gives 3×2×1=6 orderings.
Make Systematic List4.NBT.A.23STEP 3
Add the two cases
The 5- and 8-lists don't overlap — just add them.
6 + 6 = 12
Because the leading digits 5 and 8 differ, no number lands in both lists, so the counts add without overlap.
Make Systematic List4.NBT.A.2Answer
12 numbers
6 + 6 = 12
Check — these cards make 18 four-digit numbers in all. Only the 12 starting with 5 or 8 are greater than 5000.
Takeaway
Look at the biggest place first. Only cards 5 and 8 lead a number past 5000; for the rest, just list systematically and count.
- Split by the thousands digit (5 · 8)
- List the rest, none missed (6 + 6)
- Add up → 12
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested