There are 18 students involved in the class production of a play. There are 9 distinct male roles and 7 distinct female roles while the class has 9 male students and 9 female students.
In how many ways can the teacher cast the play if Jean will play the young female part only if Jovane plays the male lead?
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By Principle of Inclusion and Exclusion (PIE) . We first calculate all the possible outcome when the restriction is not considered: 9 P 5 ⋅ 9 P 7 = 2 7 4 3 3 7 2 8 0 0 Pick from 9 male students for 5 distinct role and pick from 9 female students for 7 distinct role . The we calculate the outcome in which Jovane is not male lead but Jean plays the young female ( 4 ⋅ 8 P a + 8 P 5 ) ⋅ 8 P 6 = 2 7 0 9 5 0 4 0 0 [(All cases, when Jovane is playing but not male lead) + (All cases, when Jovane is not playing)] × (Jean is young female, then pick from 8 females for 6 role) . 2 7 4 3 3 7 2 8 0 0 − 2 7 0 9 5 0 4 0 0 = 2 4 7 2 4 2 2 4 0 0