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

How many $6 -$ digit numbers can be formed from the digits, $0,1,3,5,7$ and $9$ which are divisible by $10$ and no digit is repeated?

The set $S = \left\{ {1,2,3, \ldots ,12} \right\}$ is to be partitioned into three sets $A,\,B,\, C$ of equal size . Thus $A \cup B \cup C = S$ અને $A \cap B = B \cap C = C \cap A = \emptyset $ . The number of ways to partition $S$ is

If $a, b$ and $c$ are the greatest value of $^{19} \mathrm{C}_{\mathrm{p}},^{20} \mathrm{C}_{\mathrm{q}}$ and $^{21 }\mathrm{C}_{\mathrm{r}}$ respectively, then

In how many ways $5$ speakers $S_1,S_2,S_3,S_4$ and $S_5$ can give speeches one after the other if $S_3$ wants to speak after $S_1$ & $S_2$

In a touring cricket team there are $16$ players in all including $5$ bowlers and $2$ wicket-keepers. How many teams of $11$ players from these, can be chosen, so as to include three bowlers and one wicket-keeper