Problem
Count outcomes with the multiplication rule
An outfit takes one hat, one top and one pair of pants. There are 2 hats, 3 tops and 2 pairs of pants, and any of them pair freely. Count how many different outfits are possible.
Reasoning · Grade 2-2 Applying the Multiplication Facts
Your answer
Watch out 2 + 3 + 2 = 7 is the tempting sum, but you pick one from every group at once, so the counts multiply.
How to solve
Strategy Make a Systematic List — Because each outfit is one independent choice from each group, I count by the multiplication (counting) rule: multiply the number of choices in each group. I build up the count systematically by first pairing hats with tops, then attaching pants, and a tree-style Diagram (like the lines in the figure) shows every pairing so I can trust the multiplication.
1STEP 1
Pair each hat with each top
Each hat takes any of 3 tops: 2 × 3 = 6 pairs.
2 × 3 = 6
2 hats each opening into 3 tops is 2 equal groups of 3, the meaning of 2 x 3.
3.OA.A.1Make A Systematic List2STEP 2
Attach a pair of pants to each hat-top pair
Each of the 6 takes 2 pants: 6 × 2 = 12.
6 × 2 = 12
Every one of the 6 partial outfits splits into 2 by the choice of pants, so the count doubles to 12.
3.OA.A.1Make A Systematic List3STEP 3
Read it as one multiplication
All at once: 2 × 3 × 2 = 12.
2 × 3 × 2 = 12
Multiplying the choices in each of the three groups counts all combinations at once - the counting rule.
4.OA.A.3Make A Systematic ListMultiplying the number of hats, tops and trousers together counts every complete outfit at once.
Why?
Picking a hat does not limit the tops and picking a top does not limit the trousers, so every combination really can be worn.
🧱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.
Why?
It makes no difference whether you pair hats with tops first or tops with trousers first, so the three counts may be multiplied in any grouping.
🧱Associativity of multiplicationMultiplying three numbers, grouping any two first gives the same product.
Answer
12 outfits
2 × 3 × 2 = 12
12 outfits is reasonable: it is bigger than any single group (2, 3, or 2) and equals 2 x 3 x 2. The figure's connecting lines also fan out to 12 complete hat-top-pants paths, matching the count.
Takeaway
When you pick one thing from each group, just multiply the choices: 2 x 3 x 2 = 12 outfits!
- Pair each hat with each top
- Attach a pair of pants to each hat-top pair
- Read it as one multiplication