If ${ }^{2n } C _3:{ }^{n } C _3=10: 1$, then the ratio $\left(n^2+3 n\right):\left(n^2-3 n+4\right)$ is

The value of ${}^{50}{C_4} + \sum\limits_{r = 1}^6 {^{56 - r}{C_3}} $ is

A group of students comprises of $5$ boys and $n$ girls. If the number of ways, in which a team of $3$ students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is $1750$, then $n$ is equal to

The number of words (with or without meaning) that can be formed from all the letters of the word $"LETTER"$ in which vowels never come together is

The number of onto functions $f$ from $\{1, 2, 3, …, 20\}$ only $\{1, 2, 3, …, 20\}$ such that $f(k)$ is a multiple of $3$, whenever $k$ is a multiple of $4$, is

What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these

cards are of the same colour?