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How many ways are there to select a subcommittee of 7 members from among a committte of 17?
The first member can be any one of 17 people. For each one . . .
The second member can be any one of the remaining 16 people. For each one . . .
The third member can be any one of the remaining 15 people. For each one . . .
The fourth member can be any one of the remaining 14 people. For each one . . .
The fifth member can be any one of the remaining 13 people. For each one . . .
The sixth member can be any one of the remaining 12 people. For each one . . .
The seventh member can be any one of the remaining 11 people.
Total number of ways to pick 7 people from the 17 =
(17 x 16 x 15 x 14 x 13 x 12 x 11) = 98,017,920 ways.
But that's not a useful answer. Not all of the 98 million ways produce different
subcommittees. There are a lot of different ways to pick the same 7 people and
wind up with the same group of seven.
Number of ways to select (or seat) the same 7 people = (7 x 6 x 5 x 4 x 3 x 2) = 5,040 ways.
So, among the 98 million different ways of selecting 7 people, each unique group
will show up 5,040 times.
The number of different, unique 7-member subcommittees that can be selected from
among 17 people is
98,017,920 / 5,040 = 19,448 .
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A diagram which uses overlapping circles to show the relationship among members of various sets?
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