The value of $\sum\limits_{r = 1}^{15} {{r^2}\,\left( {\frac{{^{15}{C_r}}}{{^{15}{C_{r - 1}}}}} \right)} $ is equal to

There are $m$ men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by $84,$ then the value of $m$ is

Let

$S _1=\{( i , j , k ): i , j , k \in\{1,2, \ldots, 10\}\}$

$S _2=\{( i , j ): 1 \leq i < j +2 \leq 10, i , j \in\{1,2, \ldots, 10\}\},$

$S _3=\{( i , j , k , l): 1 \leq i < j < k < l, i , j , k , l \in\{1,2, \ldots ., 10\}\}$

$S _4=\{( i , j , k , l): i , j , k$ and $l$ are distinct elements in $\{1,2, \ldots, 10\}\}$

and  If the total number of elements in the set $S _t$ is $n _z, r =1,2,3,4$, then which of the following statements is (are) TRUE?

$(A)$ $n _1=1000$   $(B)$ $n _2=44$   $(C)$ $n _3=220$   $(D)$ $\frac{ n _4}{12}=420$

If $'n'$ objects are arranged in a row then the number of ways of selecting three of these objects so that no two of them are next to each othe

Each of the $10$ letters $A,H,I,M,O,T,U,V,W$ and $X$ appears same when looked at in a mirror. They are called symmetric letters. Other letters in the alphabet are asymmetric letters. How many three letters computer passwords can be formed (no repetition allowed) with at least one symmetric letter ?

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