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Applied Combinatorics: Inclusion-exclusion formula

Bruce Springsteen, aka The Boss, has just finished a large arena show. Since he has won $20$ Grammy Awards, he decides he will toss $20$ guitar picks into the crowd as souvenirs. He has a pile of $7$ identical red picks, $7$ identical yellow picks, $7$ identical gray picks, $7$ identical green picks, and $7$ identical blue picks.

(a) Write a generating function to model the number of ways he can select the guitar picks to throw into the crowd. Use your generating function to find the number of ways to select $20$ picks.

(b) Again find the number of ways to select $20$ picks, but this time, give a solution using the inclusion/exclusion formula [Section 8.2, Theorem 1].

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