GATE EE 2020 Set 1 — Question 5
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General Aptitude → Quantitative Aptitude → Permutations & Combinations
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Question
If P, Q, R, S are four individuals, how many teams of size exceeding one can be formed, with Q as a member?
Correct answer
(C) 7
Solution
We have four individuals: P, Q, R, S.
We need to form teams of size exceeding one (i.e., size 2, 3, or 4) with Q as a mandatory member.The remaining individuals are P, R, S (3 individuals).Case 1: Team size is 2.
Since Q is already a member, we need to choose 1 more member from {P, R, S}.
Number of ways = .
Teams: {Q, P}, {Q, R}, {Q, S}Case 2: Team size is 3.
Since Q is already a member, we need to choose 2 more members from {P, R, S}.
Number of ways = .
Teams: {Q, P, R}, {Q, P, S}, {Q, R, S}Case 3: Team size is 4.
Since Q is already a member, we need to choose 3 more members from {P, R, S}.
Number of ways = .
Team: {Q, P, R, S}Total number of teams = .Alternatively, consider the set of individuals {P, R, S}. Any subset of these individuals can be combined with Q to form a team. The number of non-empty subsets of {P, R, S} is . Each of these 7 non-empty subsets, when combined with Q, forms a team of size greater than one (since Q is always present, and the subset is non-empty).Thus, the correct option is (C).
We need to form teams of size exceeding one (i.e., size 2, 3, or 4) with Q as a mandatory member.The remaining individuals are P, R, S (3 individuals).Case 1: Team size is 2.
Since Q is already a member, we need to choose 1 more member from {P, R, S}.
Number of ways = .
Teams: {Q, P}, {Q, R}, {Q, S}Case 2: Team size is 3.
Since Q is already a member, we need to choose 2 more members from {P, R, S}.
Number of ways = .
Teams: {Q, P, R}, {Q, P, S}, {Q, R, S}Case 3: Team size is 4.
Since Q is already a member, we need to choose 3 more members from {P, R, S}.
Number of ways = .
Team: {Q, P, R, S}Total number of teams = .Alternatively, consider the set of individuals {P, R, S}. Any subset of these individuals can be combined with Q to form a team. The number of non-empty subsets of {P, R, S} is . Each of these 7 non-empty subsets, when combined with Q, forms a team of size greater than one (since Q is always present, and the subset is non-empty).Thus, the correct option is (C).
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