Q: There are $5$ shirts all of different colors, $4$ pairs of pants all of different colors, and $2$ pairs of shoes with different colors. In how many ways can Amy and Bunny be dressed up with a shirt, a pair of pants, and a pair of shoes each ? ( Question from Brilliant App ).
Solution: $(5C2 \times 2!) \times (4C2 \times 2!) \times (2C2 \times 2!) = 480$
My approach:
No of permutations $= 5 \times 4 \times 2 = 40$ ways of ordering
Choosing 2 from these $40$ orderings $= 40C2 \times 2! = 1560$
Please let me know what is wrong with my approach.
Best Answer
I'd suggest to consider 2 approaches. The first one will help you to understand why your approach is not correct. Then second will help to understand how is it related to combinations and permutations.
The numbers are small. That's why first you can apply a straight forward approach, without combinations and permutations. How may choices of short do you have for Amy? It is 5. If you have chosen one color for Amy, Bunny can choose one of remaining colors. How many remain? 4. Thus, you have 5x4 pairs of shirt colors. Same about pants: Amy has 4 choices, for Bunny remain 3 choices. Thus there are 4x3 pants choices. For shoes Amy has 2 choices, but Bunny has no choice (i.e. only 1 choice) after Amy has chosen. That's why we have 2x1 choices of shoes. Thus the total number of ways is (5x4) x (4x3) x (2x1) = 480.
Now we can consider it from the point of view of combinations and permutations. We have 2 people. From 5 shirt colors we have to chose 2. How many ways are there to chose 2 out of 5? There are 5C2 ways. When we have chosen 2 shirt colors, in how many ways we can assign these colors to 2 persons? In 2! ways. If we had more colors and more persons, it would be easier to see, why we are talking about permutations here. E.g. if we had chosen 7 colors out of 20, the number of way to assign 7 colors to 7 persons would be the number of permutations, i.e. 7x6x5x4x3x2x1 = 7!. For 2 persons we have 2!. So, for shirts we have 5C2x2! For pants we have 4C2 ways to chose 2 colors out of 4. And there are 2! ways to assign them to Amy and Bunny. Thus there are 4C2x2! ways for pants. The same for shoes: 2C2x2!. The total number of ways is (5C2x2!) x (4C2x2!) x (2C2x2!) = 480.