The English alphabet has $5$ vowels and $21$ consonants. How many words with two different vowels and two different consonants can be formed from the alphabet?

The number of ways of choosing $10$ objects out of $31$ objects,of which $10$ are identical and the remaining $21$ are distinct,is

$A$ committee of $4$ members is to be formed from $4$ couples (husbands and wives). In how many ways can the committee be formed such that no couple is included in the committee?

There are $10$ bulbs in a room. Each of them can be switched on independently. In how many ways can the room be illuminated?

If ${}^n C_r$ denotes the number of combinations of $n$ things taken $r$ at a time,then the expression ${}^n C_{r+1} + {}^n C_{r-1} + 2{}^n C_r$ equals

There are $6$ different novels and $3$ different poetry books on a table. If $4$ novels and $1$ poetry book are to be selected and arranged in a row on a shelf such that the poetry book is always in the middle,then the number of such possible arrangements is