The number of arrangements of the letters of the word $SATAYPAUL$ such that no two $A$ are together and middle letter is consonant, is

Value of $r$ for which $^{15}{C_{r + 3}} = {\,^{15}}{C_{2r - 6}}$ is

For a scholarship, atmost $n$ candidates out of $2n+1$ can be selected. If the number of different ways of selection of atleast one candidate for scholarship is $63$, then maximum number of candidates that can be selected for the scholarship 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 sum $\sum\limits_{i = 0}^m {\left( {\begin{array}{*{20}{c}}{10}\\i\end{array}} \right)} \,\left( {\begin{array}{*{20}{c}}{20}\\{m - i}\end{array}} \right)\,,$ $\left( {{\rm{where}}\,\left( {\begin{array}{*{20}{c}}p\\q\end{array}} \right)\, = 0\,{\rm{if}}\,p < q} \right)$, is maximum when m is

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