If $^n{C_{12}} = {\,^n}{C_6}$, then $^n{C_2} = $
$72$
$153$
$306$
$2556$
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 value of $\sum \limits_{ r =0}^{20}{ }^{50- r } C _{6}$ is equal to
The value of $\sum\limits_{r = 0}^{n - 1} {\frac{{^n{C_r}}}{{^n{C_r} + {\,^n}{C_{r + 1}}}}} $ equals
Let $A = \left\{ {{a_1},\,{a_2},\,{a_3}.....} \right\}$ be a set containing $n$ elements. Two subsets $P$ and $Q$ of it is formed independently. The number of ways in which subsets can be formed such that $(P-Q)$ contains exactly $2$ elements, is
In an election the number of candidates is $1$ greater than the persons to be elected. If a voter can vote in $254$ ways, then the number of candidates is