The number of ordered pairs $(m, n)$,where $m, n \in \{1, 2, 3, \ldots, 50\}$,such that $6^m + 9^n$ is a multiple of $5$ is

$A$ question paper has $3$ parts $A, B, C$. Part $A$ contains $7$ questions,part $B$ contains $5$ questions,and part $C$ contains $3$ questions. If a candidate is allowed to answer not more than $4$ questions from part $A$,not more than $3$ questions from part $B$,and not more than $2$ questions from part $C$,then the number of ways in which a candidate can answer exactly $7$ questions is

Letters in the word $HULULULU$ are rearranged. The probability of all three $L$ being together is

The sum of all the four-digit numbers that can be formed using all the digits $2, 1, 2, 3$ is equal to $.......$.

The value of $\sum \limits_{r=0}^{22} {}^{22}C_{r} \cdot {}^{23}C_{r}$ is $.......$

There are $15$ stations on a train route and the train has to be stopped at exactly $5$ stations among these $15$ stations. If it stops at at least two consecutive stations,then the number of ways in which the train can be stopped is