The number of arrangements of the letters of the word $SATAYPAUL$ such that no two $A$ are together and middle letter is consonant, is
$(5!)^2$
$5!6!$
$5!4!$
$(60) × 5!$
For $2 \le r \le n,\left( {\begin{array}{*{20}{c}}n\\r\end{array}} \right) + 2\,\left( \begin{array}{l}\,\,n\\r - 1\end{array} \right)$ $ + \left( {\begin{array}{*{20}{c}}n\\{r - 2}\end{array}} \right)$ is equal to
The number of ways in which $3$ children can distribute $10$ tickets out of $15$ consecutively numbered tickets themselves such that they get consecutive blocks of $5, 3 $ and $2$ tickets is
If $^n{P_3}{ + ^n}{C_{n - 2}} = 14n$, then $n = $
The number of ways, in which $5$ girls and $7$ boys can be seated at a round table so that no two girls sit together, is
The value of $\sum\limits_{r = 1}^{15} {{r^2}\,\left( {\frac{{^{15}{C_r}}}{{^{15}{C_{r - 1}}}}} \right)} $ is equal to