The number of arrangements of the letters of the word $CALCUTTA$ is:

There are $(n + 1)$ white and $(n + 1)$ black balls,each set numbered $1$ to $(n + 1)$. The number of ways in which the balls can be arranged in a row so that the adjacent balls are of different colours is

There are $6$ men and $8$ women in a club. $A$ committee of $5$ people has to be formed. What will be the number of ways to select $5$ people if at least one woman is selected?

$A$ box contains $2$ white balls,$3$ black balls and $4$ red balls. In how many ways can $3$ balls be drawn from the box,if at least one black ball is to be included in the draw?

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

In how many ways can $6$ apples be distributed among $3$ boys,there being no restriction to the number of apples each boy may get?