Question involving permutations

Question

Assume that there are 5 unfilled roles: 3 male and 2 female. There are 6 men and 6 women, including Ann, auditioning for a part in the play. How many of these casts include Ann?

Answer 1

This could be a different interpretation of the ques.

 

 Since Ann is one of the Female cast to be choosen , we can fill the second female role out of the the available 5 females in 5 C 1 ways = 5 ways 

 

  Out of the 6 Males, the 3 positions can be filled in 5C3 ways = 20 ways.

(Note that it just chooses 3 people out of 6 for the 3 roles and does not consider how the 3 roles would be assigned to the 3 Males)

 

 so total number of ways a cast of 3 males and 2 females are choosen including Ann is 5*20 = 100.

 

Answer 2

There are 3 unfilled roles for male out of 6 men, which means there are 6 choices for the first unfilled role, 5 choices for the second, 4 choices for the third

  that is, ₆P₃ = 6 x 5 x 4 = 120

And, there are 2 unfilled roles for female including Ann out of 6 women, that means , if Ann is included, then only one unfilled role for female out of 5 women,

  which is, ₅P₁ = 5

Therefore, 120 x 5 = 600 casts