Problem
Permutations versus combinations
Four cards show 3, 1, 5 and 8. In (1) three cards are laid in a row to make a three-digit number. In (2) three cards are simply taken, order not counting. Count the possibilities for each part.
Reasoning · Grade 5-2 Counting Cases
Your answer
Watch out The parts look alike, but ignoring order makes six arrangements into one.
How to solve
Strategy Make a Systematic List — The problem itself asks for a tree diagram, and a tree is the safest kind of systematic list: it fixes the hundreds digit first, then the tens, then the ones, so nothing is repeated and nothing is missed. Counting the tips of the tree gives the answer with no formula. Then I look at the shape of the tree — 4 equal branches, each splitting 3 ways, each of those splitting 2 ways — and read the multiplication 4 times 3 times 2 straight off it, which is the second method the problem wants. For part (2) I use the easier related question 'how many ways can 3 chosen cards be arranged?', find that it is 6, and realise that the 24 numbers are just the 4 handfuls each written out in its 6 orders, so dividing by 6 removes the repeats.
1STEP 1
Build the tree, starting with the hundreds digit
Grow a tree from the hundreds digit.
3 → 1 → {5, 8} → 315, 318
Choosing places in a fixed order — hundreds, then tens, then ones — is what stops a young solver from writing the same number twice or forgetting one; the tree does the bookkeeping for you.
7.SP.C.8Draw A Diagram2STEP 2
Count the tips of the tree
The tips number 24.
6 + 6 + 6 + 6 = 24
The list is short enough to write out in full, so the answer is checked by eye rather than trusted to a rule — and the four equal groups of 6 are the clue for the next step.
7.SP.C.8Make A Systematic List3STEP 3
Read the multiplication off the shape of the tree
The tree's shape reads 4 × 3 × 2.
4 × 3 × 2 = 24
Equal groups inside equal groups is repeated multiplication, the same array thinking used for the times tables — and it lets you answer without drawing the tree once you trust it.
3.OA.A.1Make A Systematic List4STEP 4
Ask the easier question: how many ways can 3 cards be laid out?
Three cards can be ordered 6 ways.
3 × 2 × 1 = 6
Solving the small side question first — arrange 3 cards you already hold — is what makes the main question easy, and it can be done by physically shuffling three real cards.
7.SP.C.8Solve An Easier Related Problem5STEP 5
Divide away the repeats to count the handfuls
Dividing away the repeats leaves 4.
4 × 3 × 2 ÷ 6 = 24 ÷ 6 = 4
When every case has been counted the same number of times, dividing by that number removes the duplicates — the same reason you divide by 2 when counting handshakes.
3.OA.A.2Make A Systematic ListDividing away the repeats turns the count of ordered lay-outs into the count of handfuls.
Why?
Each handful of three cards shows up once for every order they could be laid in, so the ordered count is a fixed multiple of the truth.
🧱Counted twice, so halveIf every item got counted from both of its two ends, the total is exactly double the truth, so halving fixes it.
Why?
How many orders one handful has is the same for every handful, because the positions can be filled independently.
🧱Independent choices multiplyIf a first choice can go a ways and each of those leaves b ways for a second, the pair can be made a times b ways.
Answer
24, 4 ways
4 × 3 × 2 = 24, 24 ÷ 6 = 4
The 24 numbers listed by the tree are all genuinely three-digit, since none of the cards is 0, and they run from 135 up to 853, which is a sensible spread for numbers built from the digits 1, 3, 5 and 8. The tree count and the multiplication agree at 24, and they were obtained in different ways — one by writing every number out, the other by multiplying the branch counts — so the agreement is real evidence. Part (2)'s answer must be smaller than part (1)'s, because ignoring order can only merge cases together, never split them, and 4 is indeed far smaller than 24; the shrink factor 6 is exactly the number of ways 3 cards can be laid out, which is what it should be. Finally the 4 handfuls can be listed in full, and 4 also matches the 4 choices of which single card to leave behind.
Takeaway
Making a number cares about order, so multiply 4 x 3 x 2 = 24; grabbing cards does not, so divide by the 6 ways to shuffle them and get 4!
- Build the tree, starting with the hundreds digit
- Count the tips of the tree
- Read the multiplication off the shape of the tree
- Ask the easier question: how many ways can 3 cards be laid out?
- Divide away the repeats to count the handfuls