a cookie store sells 6 varieties of cookies. it has a large supply of each kind. (a) how many ways are there to select 15 cookies? (b) how many ways are there to select 15 cookies if at least three must be chocolate chip? (c) how many ways are there to select 15 cookies if at most 2 can be sugar cookies? (d) how many ways are there to select 15 cookies if at most 2 can be sugar cookies and at least 3 must be chocolate chip?

A combination is a choice made in mathematics from a group of different elements when the order of the choices is irrelevant (unlike permutations). For instance, if three fruits, such as an apple, an orange, and a pear, are supplied, there are three possible pairings of the two: an apple and a pear. Formally speaking, a k-combination of a set S is a subset of S's k unique components. In other words, two combinations are the same if and only if they have the same members. (It is not important how the individuals in each set are arranged.) The quantity of k-combinations for a set with n components

we have to select 15 Cookies from 6 varieties so we can think an n₁+n2+n3+n4 + N5 +N6=15 with each o≤ni ≤15

C([tex]\frac{20}{12}[/tex]) = 15504

NOW we have One category is fixed let K Chocolate chips are Selected in I way The A we have to (15-K) Cookies from the varieties

C([tex]\frac{19-k}{15-5}[/tex])

So for k=0, C1=3876

k=1, C2 =3060

k=2, C3=2380

So C =C1+C2+C3=9316.

So the number of ways 15504-9316= 6188.

Hence, The tottal number of ways to selct 15 cookies is 15504 and ways are there to select 15 cookies if at least three is 6188.

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