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

For $2 \le r \le n,\left( {\begin{array}{*{20}{c}}n\\r\end{array}} \right) + 2\,\left( \begin{array}{l}\,\,n\\r - 1\end{array} \right)$ $ + \left( {\begin{array}{*{20}{c}}n\\{r - 2}\end{array}} \right)$ is equal to

If $^n{C_3} + {\,^n}{C_4} > {\,^{n + 1}}{C_3},$ then

If $^n{C_r}$ denotes the number of combinations of $n$ things taken $r$ at a time, then the expression $^n{C_{r + 1}} + {\,^n}{C_{r - 1}} + \,2 \times {\,^n}{C_r}$ equals

A car will hold $2$ in the front seat and $1$ in the rear seat. If among $6$ persons $2$ can drive, then the number of ways in which the car can be filled is

A set contains $2n + 1$ elements. The number of subsets of this set containing more than $n$ elements is equal to