Question

Discrete Math:

3. (5 10+ 10) A donut shop sells 30 kinds of donuts, where donuts of the same kind are indistinguishable, and there are at least 12 donuts of each kind. In how many ways can you a. get a bag of 12 donuts? b. get a bag of 12 donuts if you want at least 3 glazed donuts and at least 4 raspberry donuts? c. get a bag of 12 donuts if you want exactly 3 glazed donuts and exactly 4 raspberry donuts?

Answer 1

a)

C(30 + 12 - 1, 12) = C(41, 12)

Since C(n, r) = C(n, n-r)

this is also equivalent to C(41, 30 - 1) = C(41, 29) = 7898654920

b)

if we want atleast 3 glazed and atleast 4 raspberry we want further 5 from the 30 kinds

C(30 + 5 - 1, 5) = C(34, 5) = C(34, 29) = 278256

c)

if we want exactly 3 glazed and exactly 4 raspberry we want further 5 from the 28 kinds

C(30 + 5 - 2 - 1, 5) = C(32, 5) = C(32, 27) = 201376