GATE | GATE-CS-2006 | Question 25

Let S = {1, 2, 3, …., m}, m>3. Let x1, x2,….xn be the subsets of S each of size 3. Define a function f from S to the set of natural numbers as, f(i) is the number of sets X_j that contain the element i. That is, f(i) = |{j|i \epsilon X_j}|.
Then, \sum_{i=1}f(i) is :
(A) 3m
(B) 3n
(C) 2m + 1
(D) 2n + 1


Answer: (B)

Explanation: First of all, number of subsets of S of size 3 is mC3 i.e. n=mC3. Now we count number of subsets in which a particular element i appears, that will be (m−1)C2, because 1 element is already known, and we have to choose 2 elements from remaining m-1 elements.

\sum\limits_{i=1}^{m} f(i) = m * ^{m-1}\mathrm{C}_2 = 3 * ^m\mathrm{C}_3 = 3n

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