Suppose a company makes n different chocolates, and they have packaged up a sele

Question

Suppose a company makes n different chocolates, and they have packaged up a selection of r of them, where r<n and each item in the selection is different. (a)

What is the total number of possible different selections?

(b) One of the chocolates is my favourite. Some of the possible selections in (a) will contain this

chocolate. How many? Some of the possible selections in (a) will not contain it. How many?

(c) Write down the definition of (m k) in terms of factorials. Now, by beginning with the left hand side, show that

Explanation / Answer

a)

As order does not matter, then there are

C(n , r) = n!/[r!(n - r)!]

ways to select.

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b)

If we fix one of them to be the favorite chocolate, then we can only choose the other r - 1 in n - 1 chocolates. Thus, there are

C(n - 1, r - 1) = (n - 1)! / [(r - 1)!(n - r)!] [ANSWER, CONTAINS THE CHOCOLATE]

ways that contain the favorite.

Those that don't contain it is the complement,

N(does not contain) = C(n, r) - C(n - 1, r - 1) [ANSWER, DOES NOT CONTAIN]

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