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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.

Jendrik Stelzner
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3 Answers3

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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.

Ross Millikan
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  • same ways dress up=$ 3^4 \times 3^4 $ ways. But how to count all possible ways? –  Aug 28 '18 at 03:33
  • No, that is all possible ways. Same ways you dress the first person then the second person has to match, so it is only the number of ways to dress the first. If you allow not using any item in each slot it should be $4$ choices instead of $3$. – Ross Millikan Aug 28 '18 at 03:36
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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.

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

Siong Thye Goh
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  • This is good. But you have to consider the other cases where the first man choose all items and the second man choose only three item or , two items or one items. So how can I include this cases? –  Aug 28 '18 at 03:31
  • You mean it is possible that the men choose not to wear something? – Siong Thye Goh Aug 28 '18 at 03:32
  • yes because it is not mandatory to wear all items according to the question –  Aug 28 '18 at 03:34
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    In that case, rather than having $3$ options for each item, view it as $4$ options, the last option means not to wear that item. – Siong Thye Goh Aug 28 '18 at 03:35
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    @MONJURALAM: I don't see where the question says they can omit any item. The way it is written I would assume they must use one of each item. If nothing is an option then Sion Thye Goh's comment is correct-they have four choices at each slot. – Ross Millikan Aug 28 '18 at 03:38