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

Determine the number of $5 -$ card combinations out of a deck of $52$ cards if each selection of $5$ cards has exactly one king.

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

A father with $8$ children takes them $3$ at a time to the Zoological gardens, as often as he can without taking the same $3$ children together more than once. The number of times each child will go to the garden is

Statement$-1:$ The number of ways of distributing $10$ identical balls in $4$ distinct boxes such that no box is empty is $^9C_3 .$

Statement$-2:$ The number of ways of choosing any $3$ places from $9$ different  places is $^9C_3 $.

The solution set of $^{10}{C_{x - 1}} > 2\;.{\;^{10}}{C_x}$ is