Problem
List all three-digit numbers from cards
Pick 3 of the number cards 2, 3, 0, 2 and use each picked card once to build a three-digit number. The hundreds digit cannot be 0. List every different three-digit number that can be built.
Reasoning · Grade 2-1 Making Numbers
Your answer
Watch out Counting arrangements blows the answer up. With two 2s, a repeated number counts once.
How to solve
Strategy Make a Systematic List — This is a 'find them all' counting task over a small finite set, so a systematic list is the natural fit. To stay organized I first reduce the work by choosing which 3 cards to use (three possible groups), then list the numbers from each group by fixing the hundreds digit. Fixing the hundreds digit turns one hard listing into a few easier sub-listings.
1STEP 1
Decide which 3 cards to keep
Leaving one card out gives just three groups: 2·3·0, 2·2·0, 2·2·3.
Sorting the work by 'which three digits' first means I never mix groups and never miss one, using only the place-value idea that a 3-digit number is built from three digits.
2.NBT.A.1Solve An Easier Related ProblemSorting the work by which three cards are kept splits every possible number into groups that never overlap and leave nothing out.
Why?
A number belongs to exactly one group because it is built from exactly one set of three cards, so the group counts can simply be added at the end.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
Why?
Inside a group with two identical cards, swapping those two cards produces the very same numeral, so two arrangements can name one number.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
2STEP 2
List numbers from the digits 2, 3, 0
In 2·3·0 the 0 cannot lead, giving 203, 230, 302, 320.
{2,3,0} → 203, 230, 302, 320
Fixing the hundreds digit and then ordering the two leftovers is the easy, no-skip way to write every arrangement.
2.NBT.A.3Make A Systematic List3STEP 3
List numbers from the digits 2, 2, 0
In 2·2·0 only 2 can lead, giving 202, 220.
{2,2,0} → 202, 220
The two identical 2-cards mean swapping them changes nothing, so this group makes only the two distinct numbers.
2.NBT.A.3Make A Systematic List4STEP 4
List numbers from the digits 2, 2, 3
In 2·2·3 the repeated 2 leaves only 223, 232, 322.
{2,2,3} → 223, 232, 322
Again fixing the hundreds digit makes the listing easy, and the repeated 2 collapses the would-be 322/322 pair into a single number.
2.NBT.A.3Make A Systematic List5STEP 5
Combine and remove duplicates
Together they make nine numbers with no repeats, the smallest 202.
202, 203, 220, 223, 230, 232, 302, 320, 322
Sorting the combined list from smallest to largest makes it obvious that none are missing or repeated.
2.NBT.A.4Make A Systematic ListAnswer
202, 203, 220, 223, 230, 232, 302, 320, 322 numbers (9)
Every number is three digits with no leading zero, and each uses only digits the cards allow (at most two 2's, one 3, one 0). The largest is 322 and the smallest is 202, both sensible for these cards. Nine distinct numbers is reasonable: with all-different digits there could be up to a handful per group, and the repeated 2 trims the count.
Takeaway
Sort first by which cards you keep, then by the hundreds digit, and you can list every number without missing one — pure Grade 2 place-value thinking!
- Decide which 3 cards to keep
- List numbers from the digits 2, 3, 0
- List numbers from the digits 2, 2, 0
- List numbers from the digits 2, 2, 3
- Combine and remove duplicates