In a club election,the number of contestants is one more than the maximum number of candidates for which a voter can vote. If the total number of ways in which a voter can vote is $62$,then the number of candidates is:

In how many ways can $5$ distinct balls be distributed among $3$ persons such that each person receives at least one ball?

$A$ debate club consists of $6$ girls and $4$ boys. $A$ team of $4$ members is to be selected from this club,including the selection of a captain (from among these $4$ members) for the team. If the team has to include at most one boy,then the number of ways of selecting the team is

The number of all $3$-digit numbers $abc$ (in base $10$) for which $(a \times b \times c) + (a \times b) + (b \times c) + (c \times a) + a + b + c = 29$ is:

The value of ${}^{47}C_4 + \sum_{j=1}^5 {}^{(52-j)}C_3$ is

From a collection of $20$ consecutive natural numbers, four are selected such that they are not consecutive. The number of such selections is