The number of ways in which $35$ apples can be distributed among $3$ boys so that each can have any number of apples,is

$A$ man has $10$ friends. In how many ways can he invite one or more of them to a party?

$A = \{ x_1, x_2, x_3, x_4 \}; \, B = \{ y_1, y_2, y_3, y_4 \}.$ $A$ function is defined from set $A$ to set $B.$ The number of one-one functions such that $f(x_i) \neq y_i$ for $i = 1, 2, 3, 4$ is equal to:

$^{14}C_4 + \sum_{j=1}^4 {^{18-j}C_3}$ is equal to

The number of integers greater than a million that can be formed using the digits $2, 3, 0, 3, 4, 2, 3$ is:

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