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Combinations

When the order in which objects are chosen is not important, as it is in a permutation, the arrangement is called a combination. The combination of n things taken r at a time is written as C(n, r) or nC r . A formula for finding the number of combinations of n objects taken r at a time is given by the following:

Note that this is the same as the binomial coefficient formula.

Booties Stiletto Leather Zip Martin Pointed Honeystore Toe Side Ruched Boots Heel Women's Brown Example 1

A class has 16 students. How many groups consisting of four students can be formed?

This is a combinations problem, since the order in which the students in a group are chosen is not important. In this problem, n = 16 and r = 4.

There are 1820 different ways to form the groups.

Example 2

A committee in Congress consists of nine Republicans and six Democrats. In how many ways can a subcommittee be chosen if it must contain four Republicans and three Democrats?

There are C(9, 4) ways of choosing the four Republicans out of the nine Republicans. There are C(6, 3) ways of choosing the three Democrats out of the six Democrats. By the multiplication principle of events, there are C(9, 4) · C(6, 3) ways of choosing the subcommittee.

Booties Boots Heel Honeystore Pointed Brown Ruched Zip Side Leather Toe Martin Stiletto Women's There are 2520 different possible subcommittees.

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