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Let S=\{1,2,3, \ldots, n\}. If X denote the set of all subsets of S containing exactly two elements, then the value of \sum_{A \in X}(\min A) is given by 

Option: 1

{ }^{n+1} C_3


Option: 2

{ }^n C_3


Option: 3

\frac{n\left(n^2-1\right)}{6}


Option: 4

\frac{(n-1)^n}{6}


Answers (1)

best_answer

We have (n-1) elements subsets of S having 1 as least element, (n-2) elements subsets of S having 2 as least element, (n-3) elements subsets of S having 3 as least element, ......1 elements subset of S having (n-1) as least element.

Thus, \begin{aligned} & \sum_{A \in X}(\min A)=\sum_{r=1}^{n-1} r(n-r)=n \sum_{r=1}^{n-1} r-\sum_{r=1}^{n-1} r^2 \\ \\& \quad=\frac{n(n-1) n}{2}-\frac{(n-1) n(2 n-1)}{6} \end{aligned}

      \begin{aligned} & =\frac{(n-1) n}{6}(3 n-2 n+1) \\ \\& =\frac{(n+1) n(n-1)}{1 \cdot 2 \cdot 3}={ }^{n+1} C_3=\frac{n\left(n^2-1\right)}{6} \end{aligned}

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