The value of $\sum_{r=0}^{20} {}^{50-r}C_{6}$ is equal to

Consider $4$ boxes,where each box contains $3$ red balls and $2$ blue balls. Assume that all $20$ balls are distinct. In how many different ways can $10$ balls be chosen from these $4$ boxes so that from each box at least one red ball and one blue ball are chosen?

The number of seven-digit numbers that can be formed by using the digits $1, 2, 3, 5,$ and $7$ such that each digit is used at least once is:

There are $n$ white and $n$ black balls marked $1, 2, 3, \ldots, n$. The number of ways in which we can arrange these balls in a row so that neighbouring balls are of different colours is

Five balls of different colors are to be placed in three boxes of different sizes,where each box can hold all five balls. In how many ways can the balls be placed such that no box remains empty?

If $n$ is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero,then $n$ is equal to: