平成21年度秋期 ITパスポート試験問46

This question's explanation and your answers so far.

問46

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あるプロジェクトの関係者6人が,それぞれ1対1で情報の伝達を行う必要があるとき,情報の伝達を行うために必要な経路の数は少なくとも幾つになるか。

出典　平成21年度秋期 ITパスポート試験問46

解説

問 46

Incorrectア
6 is simply the number of people. The number of people and the number of paths (pairs) are different things; 6 paths would only correspond to arranging the 6 people in a ring and connecting neighbours, which does not connect everyone one-to-one.
Incorrectイ
9 does not come out of any way of counting. Both 6C2 and 6×5÷2 give 15, so 9 has no basis.
ウ
正解 For 6 project members to exchange information one-to-one with each other, one path is needed for every pair chosen from the 6 people. That number is the combination 6C2 = (6×5)÷(2×1) = 15. You can check it another way: each person needs a path to the other 5, giving 6×5 = 30, but the path from A to B and the path from B to A are the same single path, so divide by 2: 30÷2 = 15 paths. Therefore 15, choice ウ, is correct.
Incorrectエ
30 is 6 people × the other 5 people, that is the permutation 6P2, which counts A-to-B and B-to-A separately. A path is bidirectional and counts as one, so it must be divided by 2; 30 is twice the correct value.

ここがポイント

The number of paths (communication paths) connecting n people one-to-one is the combination nC2 = n(n−1)÷2. For n=6 that is 6×5÷2 = 15.
Because a path is bidirectional and counts as one, do not use the permutation n(n−1) (= 30) as the answer; always divide by 2. In project management, note that communication paths grow rapidly as the number of people increases.

用語

解答問46ウ