Mathematics - Permutation & Combination Question with Solution | TestHub

There are exactly ( $n-1$ ) sub sets of S containing two elements having 1 as least element; exactly ( $\mathrm{n}-2$ ) subsets of S having 2 as least element and so on.

$$\begin{array}{rlr}& \text{ Thus }\sum _{A\in X}\min (A)=(1)(n-1)+2(n-2)+\dots +(n-1)(1)& \end{array}$$

$$=\sum _{r=1}^{n-1}r(n-r)=n+1_{C_3}$$