# A play needs three protagonists (a man, a woman, and a girl), two antagonists (a man and a woman), and two supporting roles (a man and a boy).

A play needs three protagonists (a man, a woman, and a girl), two antagonists (a man and a woman), and two supporting roles (a man and a boy).

Six men, six women, three boys, and four girls attended the auditions. The number of ways the roles can be cast is:

woman can be protagonists or antagonists

six women are there

2 women can be selected in ⁶C₂ = 15 ways and can be assigned role in 2! = 2 ways

Hence total ways role to woman can be assigned = 15 x 2 = 30

Man can be protagonists , antagonists , supporting role

six men are there

3 men can be selected in ⁶C₃ =20 ways and can be assigned role in 3! = 6 ways

Hence total ways role to woman can be assigned =20 x 6 = 120

Girl can be protagonists

There are 4 girls

Hence 4 ways

Boy can be supporting roles

There are 3 boys

Hence 3 ways

Total Ways = 30 x 120 x 4 x 3

= 43,200

The number of ways the roles can be cast is: 43200