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

Options

A. `.^(n+1)C_(3)`
B. `.^(n)C_(3)`
C. `(n(n^(2)-1))/(6)`
D. `(n(n^(2)-3n+2))/(6)`

Answer

A, C

Explanation

`underset(A in X)(sum)` minA
`underset(r=1)overset(n-1)(sum)r(n-r)=n underset(r=1)overset(n-1)(sum)r-underset(r=1)overset(n-1)(sum)r^(2)`
`=(n(n-1)n)/(2)-((n-1)n(2n-1))/(6)`
`=((n+1)*n*(n-1))/(1*2*3)=.^(n+1)C_(3)=(n(n^(2)-1))/(6)`