The number of even numbers greater than $1000000$ that can be formed using all the digits $1, 2, 0, 2, 4, 2, 4$ is

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

Let $S$ denote the set of $4$-digit numbers $abcd$ such that $a > b > c > d$ and $P$ denote the set of $5$-digit numbers having the product of its digits equal to $20$. Then $n(S) + n(P)$ is equal to:

In how many ways can $3n$ distinct objects be distributed equally among $3$ persons?

An eight-digit number divisible by $9$ is to be formed using digits from $0$ to $9$ without repeating the digits. The number of ways in which this can be done is: (in $(7!)$)

Ten students are seated at random in a row. The probability that two particular students are not seated side by side is