The teacher wants to arrange $5$ students on the platform such that the boy $B_1$ occupies the second position and the girls $G_1$ and $G_2$ are always adjacent to each other. Then,the number of such arrangements is

If ${}^9P_5 + 5 \cdot {}^9P_4 = {}^{10}P_r$, then the value of $r$ is:

The number of ways in which all the letters of the word $COMBINATION$ can be arranged so that the vowels always come together is:

For $n = 1, 2, 3, . . . , 50$,let $A = \{ a_n \mid a_n = \begin{cases} (-1)^{\frac{n}{2}} (\frac{n}{2}), & \text{if } n \text{ is even} \\ (-1)^{\frac{n-1}{2}} (\frac{n-1}{2}), & \text{if } n \text{ is odd} \end{cases} \}$ and $B$ be the set of all distinct elements of $A$. The number of permutations of all the elements of set $B$ such that even integers are in increasing order is:

How many words can be made from the letters of the word $DELHI$,if $L$ comes in the middle in every word?

All the letters of the word $LETTER$ are arranged in all possible ways and the words (with or without meaning) thus formed are arranged in dictionary order. Then the rank of the word $TETLER$ is