Question

A cupboard contains 5 pairs of shoes, each of a different style. How many ways are there to select 4 of the shoes from the cupboard so that the selection contains exactly one matching pair? Note that each pair of shoes consists of distinguishable left and right foot.

Answer 1

A1A2 B1B2 C1C2 D1D2 E1E2

Firstly select any single pair ->C(5,1)
then you have to select 2 more shoes from the remaining 8.
1st choice can be anything ->C(8,1)
after selecting 1st, we have only 6 shoes left to chose from ->C(6,1)

hence final answer will be C(5,1) * C(8,1) * C(6,1)

Answer 2

There are 10 ways of choosing the first shoe. There is one matching shoe among the remaining 9 shoes and this matching shoe can be combined in 8C2 ways with the 8 non-matching shoes. Therefore the number of ways to select 4 shoes containing exactly one matching pair is:
\[10\times 8C2=10\times \frac{8!}{2!6!}=\frac{10\times 8\times 7}{2}\]

Answer 3

@kropot72 In 8C2 , you're also likely to get another pair of matching shoes, isnt it ?

Answer 4

@shubhamsrg You are correct. There are 4 ways of getting a matching pair among the remaining 8 shoes that do not match the first shoe selected.
Therefore the number of ways to select 4 shoes containing exactly one matching pair is:
\[10\times (8C2-4)=10\times (\frac{8!}{2!6!}-4)=10\times \frac{(8\times 7)-8}{2}=\frac{10\times 8\times 6}{2}\]
which is the same as your result.