How many ways can they dress so that are not both dressed exactly the same?

Question

If two man have the same $ 3 \ $ pairs of shoes, the same $3 \ $ pairs of pants, the same $ 3 \ $ shirts, and the same $ \ 3$ sweaters. How many ways can they dress so that are not both dressed exactly the same?

Answer:

Suppose one man choose to wear all $ \ 4 $ items but the second man reject anyone item and choosen the rest three item , then they can be dress up not exactly same with $$ \underbrace{(6 \times 6 \times 6 \times 6)}_{\text{for the first man}} \times \underbrace{(5 \times 5 \times 5)}_{\text{for the $2$nd man}} $$ ways. But since there are $4$ types of item , the first man can choose the fourth item in $4$ ways.

Thus the total ways to dress up the two mans is $$ 4 \times \underbrace{(6 \times 6 \times 6 \times 6)}_{\text{for the first man}} \times \underbrace{(5 \times 5 \times 5)}_{\text{for the $2$nd man}}. $$

I need help solving this problem.

Answer 1

I'm not sure at all where you're getting the $6$'s from in your answer, but here's a hint.

Calculate the number of ways that they can dress exactly the same, and subtract that number from the total number of ways they can both dress.

How many ways can one of the men dress?

Spoiler:

One of the men can dress $3^4 = 81$ ways. This is also the number of ways that the two men can dress the same. The total number of ways they can dress is $81^2 = 6561$ ways. The answer you want is the difference of these two numbers.

Answer 2

HInt: How many ways to dress them both if you ignore the restriction? Subtract the number of ways to dress them that are identical, which is just the number of ways to dress the first as the second has to match.

Answer 3

I am going to assume that the first person has no access to second person's clothes.

The first person can dress in $3^4$ ways.

The second person just have to avoid it.

Hence $$3^4 ( 3^4 -1)$$