If $n$ and $r$ are two positive integers such that $n \ge r,$ then $^nC_{r-1} + ^nC_r = $

The number of ways that $8$ beads of different colours can be strung as a necklace is:

The number of permutations of $n$ things taken $r$ at a time,when $p$ things are always included,is

Six married couples decide to form a committee of $6$ persons. The total number of ways to form the committee such that it contains no couples is equal to:

The sum of all $4$ digit numbers that can be formed by using the digits $2, 4, 6, 8$ (repetition of digits not allowed) is

The number of arrangements that can be formed from the letters $a, b, c, d, e, f$ taken $3$ at a time without repetition and each arrangement containing at least one vowel,is