問42
頻出度★★★
プロジェクトメンバ A, B, C の3者間で直接一対一でコミュニケーションをする場合には, A～B 間, B～C 間, C～A 間の三つの伝達経路が存在する。7人でコミュニケーションする場合の伝達経路は最大で幾つになるか。
平成24年度 秋期 ITパスポート試験 問42
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This question's explanation and your answers so far.
解説
問 42
Incorrectア
7 is simply the number of people n, not the number of paths. If the 7 people were arranged in a ring (circle) and only adjacent people were connected, the count would indeed be 7, but in the three-person example the question counts every combination of two people — between A and B, between B and C, and between C and A — as a path, and there is no condition anywhere limiting connections to adjacent people. Checking with the question's example, counting only adjacent pairs also gives 3 for three people, so it happens to coincide; but for 7 people there are 21 combinations connecting everyone with everyone else, so 7 is not enough. This is a wrong answer that confuses the number of people with the number of paths.
Incorrectイ
14 is 7×2, which corresponds to assuming that each person has 2 paths. However, the partners for one-to-one direct communication are everyone other than oneself, that is (7-1) = 6 people, not 2. Applying the same idea to the three-person example in the question gives 3×2 = 6, which does not match the actual 3 paths, confirming that this way of counting is wrong. This is a wrong answer that mistakes the (n-1) part of the formula n×(n-1)÷2.
- ウ
正解 The correct answer is ウ“21”. In one-to-one direct communication, one path is created for each combination of choosing two different people. Therefore, for n people the number of paths is given by the combination formula nC2 = n×(n-1)÷2. This formula can be derived as follows. First, looking at any one person, that person has a path with each of the other (n-1) people, so counting this for everyone gives n×(n-1) paths. However, the path between A and B and the path between B and A are the same single path counted twice from both ends, so you must divide by 2 to remove the duplication. Checking with the three-person example given in the question, 3×(3-1)÷2 = 3, which matches the statement that there are three paths — between A and B, between B and C, and between C and A — confirming that this reasoning matches the intent of the question. Substituting n=7 into the same formula gives 7×(7-1)÷2 = 7×6÷2 = 42÷2 = 21, so there are at most 21 communication paths. The correct answer is therefore ウ. Note that as the number of people grows, the number of paths increases rapidly, roughly in proportion to the square of the number of people (3 people → 3 paths, 7 people → 21 paths, 10 people → 45 paths). For this reason, in Project Communications Management it is important to use measures such as designating a point of contact or consolidating information into meeting bodies and information-sharing platforms, so as to curb the missed messages and misunderstandings caused by the growth in the number of paths.
Incorrectエ
42 is 7×(7-1) = 7×6, the number before dividing by 2. This is the count obtained when “from A to B” and “from B to A” are counted separately, that is, when direction is distinguished (the permutation 7P2); it counts the same single path twice from both ends. The paths in the question are counts of undirected “between” pairs, such as between A and B, between B and C, and between C and A, so you must divide by 2 to remove the duplication. Checking with the three-person example gives 3×2 = 6, exactly twice the actual 3 paths, which shows that this is a wrong answer that forgets to divide by 2.
ここがポイント
- The number of communication paths (communication channels) is counted with combinations — when n people communicate directly one-to-one with each other, the number of paths is nC2 = n×(n-1)÷2. Since n×(n-1) counts A-to-B and B-to-A twice, you must always divide by 2 to remove the duplication. 3 people = 3 paths, 7 people = 21 paths, 10 people = 45 paths: the number of paths grows rapidly, roughly with the square of the number of people.
- The thinking behind Project Communications Management — when the number of paths grows rapidly, missed messages, misunderstandings, and coordination costs all increase, so communication becomes harder as more members are added. Organizing the paths — designating a point of contact (a representative), splitting the team into smaller groups, and consolidating information into meeting bodies or information-sharing tools — is an important consideration at the planning stage.
用語