GATE 2020 EE GA Q.5: solution and answer

GATE 2020 · EEGA Q.5 · 1 mark · MCQ

If $P, Q, R, S$ are four individuals, how many teams of size exceeding one can be formed, with $Q$ as a member?

(A)
5
(B)
6
(C)
7
(D)
8

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Answer and solution

ANSWER

7 teams (option C).

Official answer: CSource: Official answer key, GATE 2020 EE (IIT Delhi)

$Q$ is in every team, so a team is decided only by which of the other three people ( $P$ , $R$ , $S$ ) join $Q$ .
"Size exceeding one" means at least one of $P, R, S$ must join $Q$ .
Each of $P, R, S$ is either in or out, independently, which gives $2^{3} = 8$ subsets. Remove the empty subset, because that leaves $Q$ alone in a team of size 1: $2^{3} - 1 = 7$ .
Check by team size, using combinations: size 2 gives $(1 3) = 3$ ( $QP, QR, QS$ ), size 3 gives $(2 3) = 3$ ( $QP R, QP S, QR S$ ), size 4 gives $(3 3) = 1$ ( $P QR S$ ).
Total $= 3 + 3 + 1 = 7$ , which is option (C).

KEY IDEA

Fix $Q$ in the team, then count the subsets of the other three people. There are $2^{3} = 8$ of them. Drop the empty one, because a team of $Q$ alone has size 1. That leaves $7$ .

WATCH OUT FOR

Two slips are common. The first is keeping the empty subset, which counts $Q$ alone as a team and gives 8 (option D). The second is leaving out the full team $P QR S$ , which gives 6 (option B). A third error is counting every team of size 2 or more from all four people, which is $2^{4} - 4 - 1 = 11$ , and ignoring the condition that $Q$ must be a member.

From the GATE 2020 EE paper. Every question of this paper · Download the full paper · All 58 Numerical questions, with notes

GATE 2020 EE GA Q.5: solution and answer

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