Question

Charlie has 4 pairs of shoes, 12 shirts, 5 pairs of pants, and 3 watches. How many days could he go without wearing the same combination of these four items?

Answer 1

We have four sets

 

4 shoes

12 shirts

5 pants

3 watches

 

But let's start with a simpler example - say we're only interested in shirts and pants. I suppose you could wear any one of your pants every day while you cycle through each of your 12 shirts. Then you can do that again for the next pair of pants, and again for the next. You would have 12 outfits for each of your 5 pairs of pants, so the number of these outfits would be 12×5

 

Once you start including in shoes, you could wear one of your pairs of shoes while you cycle through all your pants/shirts outfits. Then you can wear them all again for your next pair of shoes, and again for each pair of shoes. This means that the number of these outfits would be 4×12×5

I hope we see that if we cycle through the total combinations from four sets, we will just multiply the number of elements in each set, so 4×12×5×3

 

720 days

Answer 2

Charlie can go 720 days without wearing the same combination of the 4 items.

Charlie has 4 pairs of shoes and 12 shirts, hence he can wear them in 4*12 ways.

Along with this, he also has 5 pairs of pants. Hence, he can combine them in 4*12*5 ways.

He also has 3 watches. Therefore, he can combine them in 4*12*5*3 ways

=720