The factorial function occurs often in probability and statistics. For a non-negative integer n, the factorial is denoted n! (which is read "n factorial") and is defined as follows: First, 0! is defined to be 1. Next, if n is 1 or larger then n! means n(n − 1)(n − 2)...3 ✕ 2 ✕ 1. Thus 3! = 3 ✕ 2 ✕ 1 = 6.

C =

n!

k!(n − k)!

if k is not greater than n.

(a) How many different groups of 3 people could be selected from a group of 20 to go on a trip?

To determine how many different groups of 3 people could be selected from a group of 20 to go on a trip, we first note that once we pick the 3 people, the order in which we picked them does not matter. For example, picking Alex, Bob, and Cheryl to go would be the same as picking Cheryl, Alex, and Bob. If order did matter, then the answer would simply be 20*19*18. There are 20 choices for the first pick, then only 19 for the second (because we have already picked one), and 18 for the third. But because order does not matter, this number is too big. So we must ask, for each group of 3 people (e.g. Alex, Bob, and Cheryl) how many ways can we reorder them? The answer is 3*2*1=3!=6. So we take 20*19*18 and divide by 6 to arrive at our answer of 1140.

This is an example of a combination problem (where order does not matter) as opposed to a permutation problem (where order does matter). In general there is a formula to determine number of combinations. In our case we want to compute the number "20 choose 3" which is given by the formula in the problem. 20 choose 3 = 20!/(3! * 17!). 20!/17! = 20*19*18, so we see that we get the same answer or 1140.

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Sep 22nd, 2015

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