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Question

Difficulty: EasyPermutations and Combinations

A board of directors consists of 77 members. In how many different ways can a subcommittee of 33 members be chosen from the board?

Answer: 35

Answer

35 different subcommittees can be chosen.
The total number of ways to choose a subcommittee of 3 members from a group of 7 members when order does not matter is given by the combination formula (73)=7×6×53×2×1=35\binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35.

Step-by-Step Solution

1
Determine whether the selection requires permutations or combinations.
Since selecting members A, B, and C forms the exact same committee as selecting B, C, and A, the order of selection does not matter. Therefore, this is a combination problem.
Committees are unordered groups.
2
Calculate the number of combinations of 7 items taken 3 at a time using (73)=7!3!(7−3)!\binom{7}{3} = \frac{7!}{3!(7-3)!}.
(73)=7×6×53×2×1=35\binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35.
Simplify the factorial expression by canceling out common terms.

Key Concept

Combinations (Selection Without Regard to Order)
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Topics

Permutations and Combinations

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