If $\frac{n!}{2!(n-2)!}$ and $\frac{n!}{4!(n-4)!}$ are in the ratio $2:1$,then $n=$

If the number of $5$-element subsets of the set $A = \{a_1, a_2, \dots, a_{20}\}$ containing $20$ distinct elements is $k$ times the number of $5$-element subsets containing the element $a_4$,then what is the value of $k$?

If ${}^{2n}C_3 : {}^{n}C_3 = 10 : 1$,then the ratio $(n^2 + 3n) : (n^2 - 3n + 4)$ is

The number of integers between $10$ and $10,000$ such that in every integer every digit is greater than its immediate preceding digit,is

In a touring cricket team,there are $16$ players in all,including $5$ bowlers and $2$ wicket-keepers. How many teams of $11$ players can be chosen from these,such that the team includes exactly $3$ bowlers and $1$ wicket-keeper?

In how many ways can $5$ letters be selected from the letters of the word $INDEPENDENT$?