A donut shop has six different kinds of donuts: glazed, plain, chocolate, cake,

Question

A donut shop has six different kinds of donuts: glazed, plain, chocolate, cake, lemon filled, and powdered. Suppose that you go to the shop to buy three dozen donuts for the class and that you must buy at least 3 glazed donuts, at least 2 chocolate donuts, and exactly 5 cake donuts in order to satisfy certain members of the class (your classmates are picky...). The shop has an unlimited supply of each type of donut, EXCEPT they only have 2 plain donuts left (there was a run on them just before you arrived...). Given these restrictions, how many ways are there for you to buy 3 dozens donuts? (SHOW WORK)

Explanation / Answer

There are 10 donuts ( 3 - glazed, 2 - chocolate and 5 - cake) has to be there in 3 dozens i.e. 36 donuts. Now there are only 26 donuts that are need to be chosen from 6 different types of donuts.

Another condition is that there are only 2 plain donuts available in the shop. In our selection we can have 0, 1 or 2 of them.

Let a, b, c, d and e are the number of different kinds of donuts except plain

If we select 0 plain donuts,

a + b + c + d + e = 26

number of integer solution to this equation is (n+k-1)C(k-1) = (26+5-1)C(5-1) = 30C4

If we select 1 plain donut,

a + b + c + d + e = 26 -1

a + b + c + d + e = 25

number of integer solution to this equation is (n+k-1)C(k-1) = (25+5-1)C(5-1) = 29C4

similarly, for 2 plain donuts, solution would be 28C4

Total number of ways to select 6 different types of donuts = 30C4 + 29C4 + 28C4 = 27405 + 23751 + 20475 = 71631

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