Sooooooo many ways!!!

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?


The answer is 2472422400.

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1 solution

Kevin Xu
Sep 18, 2019

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 = 2743372800 9P5 \cdot 9P7 = 2743372800 \\ 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 = 270950400 (4\cdot8Pa+8P5)\cdot 8P6 = 270950400 \\ [(All cases, when Jovane is playing but not male lead) + (All cases, when Jovane is not playing)] × \times (Jean is young female, then pick from 8 females for 6 role) \\ . .\\ 2743372800 270950400 = 2472422400 2743372800 - 270950400 = 2472422400

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