$n$ balls each of weight $w$ when weighted in pairs the sum of the weights of all the possible pairs is $120$ when they are weighed in triplets the sum of the weights comes out to be $480$ for all possible triplets, then $n$ is

A group of $9$ students, $s 1, s 2, \ldots, s 9$, is to be divided to form three teams $X, Y$ and, $Z$ of sizes $2,3$ , and $4$, respectively. Suppose that $s_1$ cannot be selected for the team $X$, and $s_2$ cannot be selected for the team $Y$. Then the number of ways to form such teams, is. . . .

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 :

What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these

four cards are of the same suit,

If $^n{C_{r - 1}} = 36,{\;^n}{C_r} = 84$ and $^n{C_{r + 1}} = 126$, then the value of $r$ is

On the occasion of Deepawali festival each student of a class sends greeting cards to the others. If there are $20$ students in the class, then the total number of greeting cards exchanged by the students is