$\binom{n}{r+1} + 2\binom{n}{r} + \binom{n}{r-1} = \dots$

The total number of all those $3$-digit numbers in which the sum of all the digits in each of them is $10$,is

For a scholarship,at most $n$ candidates out of $2n+1$ can be selected. If the number of different ways of selecting at least one candidate for the scholarship is $63$,then the maximum number of candidates that can be selected for the scholarship is -

The number of elements in the set $\{x \in N : ^{20-2x}C_{x-3} \in N\}$ is

If $1 \times 1! + 2 \times 2! + 3 \times 3! + \ldots + n \times n! = 11! - 1$,then the maximum value of ${}^n C_r$ is

What is the number of ways in which an examiner can assign $10$ marks to $4$ questions, giving not less than $2$ marks to any question?