If $^{n + 1}{C_3} = 2{\,^n}{C_2},$ then $n =$

How many words can be formed by taking $3$ consonants and $2$ vowels out of $5$ consonants and $4$ vowels?

$A$ bag contains balls of two colours,$3$ black and $3$ white. What is the minimum number of balls which must be drawn from the bag,without looking,so that among these there are two of the same colour?

If $\alpha = {^m}{C_2}$,then ${^\alpha}{C_2}$ is equal to

How many arrangements can be made of the letters of the word '$ARRANGEMENT$'?

For $2 \le r \le n$,the expression $\binom{n}{r} + 2\binom{n}{r-1} + \binom{n}{r-2}$ is equal to: