Q:

Nine wolves, eight female and one male, are to be released into the wild three at a time. If the male wolf is to be in the first released group and order does not matter, in how many ways can the first group of three wolves be formed?

Accepted Solution

A:
We have been given that Nine wolves, eight female and one male, are to be released into the wild three at a time.We need to choose 1 male wolf out of 1 wolf male. So  we can choose one wolf as: [tex]_{1}^{1}\textrm{C}[/tex][tex]\frac{1!}{1!(1-1)!}[/tex][tex]\frac{1!}{1!\cdot 0!}[/tex][tex]\frac{1}{1\cdot 1}=1[/tex]We can choose 1 male wolf in only 1 way.Since the female wolfs are identical (order doesn't matter), so we will use combinations.We can choose 2 female wolves out of 8 as:[tex]_{2}^{8}\textrm{C}[/tex][tex]\frac{8!}{2!(8-2)!}[/tex]   [tex]\frac{8\cdot 7\cdot 6!}{2\cdot 1\cdot 6!}=\frac{8\cdot7}{2}=28[/tex]Therefore, we can choose 2 female wolves out of 8 female wolves in 28 ways.To find number of ways in which the first group of three wolves can be formed we will multiply the ways of choosing 1 male wolf and 2 female wolves.[tex]\text{Number of ways of forming first group of three wolves}=1\times 28=28[/tex]Therefore, the first group of three wolves can be formed in 28 ways.