If for some $m, n$,${ }^6 C_{m}+2({ }^6 C_{m+1})+{ }^6 C_{m+2} > { }^8 C_3$ and ${ }^{n-1} P_3 : { }^n P_4 = 1 : 8$,then ${ }^n P_{m+1} + { }^{n+1} C_m$ is equal to

$A$ man $P$ has $7$ friends,$4$ of them are ladies and $3$ are men. His wife $Q$ also has $7$ friends,$3$ of them are ladies and $4$ are men. Assume $P$ and $Q$ have no common friends. Then the total number of ways in which $P$ and $Q$ together can throw a party inviting $3$ ladies and $3$ men,so that $3$ friends of each of $P$ and $Q$ are in this party,is . . . . . . .

Let $m = (9n^2 + 54n + 80)(9n^2 + 45n + 54)(9n^2 + 36n + 35)$. The greatest positive integer which divides $m$ for all positive integers $n$ is:

The total number of selections of the letters in the phrase "ned needs nineteen nets" is:

$A$ man has $7$ relatives,$4$ of them are ladies and $3$ are gents; his wife has $7$ other relatives,$3$ of them are ladies and $4$ are gents. The number of ways they can invite $3$ ladies and $3$ gents to a party such that there are $3$ of the man's relatives and $3$ of the wife's relatives,is:

The remainder of $n^4-2n^3-n^2+2n-26$ when divided by $24$ is: