If $C(n, 7) = C(n, 5)$,find $n$.

The number of triangles that can be formed by choosing the vertices from a set of $12$ points,$7$ of which lie on the same straight line,is

$n$-digit numbers are formed using only three digits $2, 5$,and $7$. The smallest value of $n$ for which $900$ such distinct numbers can be formed is:

We wish to select $6$ persons from $8,$ but if the person $A$ is chosen,then $B$ must be chosen too. In how many ways can the selection be made?

Let $A = \{a_1, a_2, a_3, ..., a_n\}$ be a set containing $n$ elements. Two subsets $P$ and $Q$ of $A$ are formed independently. The number of ways in which these subsets can be formed such that $(P - Q)$ contains exactly $2$ elements is:

If $a_n = \sum_{r = 0}^n \frac{1}{^nC_r}$,then $\sum_{r = 0}^n \frac{r}{^nC_r}$ equals