The value of $\sum\limits_{r = 1}^{15} {{r^2} \left( \frac{^{15}C_r}{^{15}C_{r - 1}} \right)}$ is equal to

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

The number of integers greater than $6000$ that can be formed using the digits $3, 5, 6, 7,$ and $8$ without repetition is:

${ }^{34} C_5+\sum_{r=0}^4{ }^{(38-r)} C_4=$

If $n$ is an integer with $0 \leq n \leq 11$,then the minimum value of $n!(11-n)!$ is attained when a value of $n$ equals to

$A$ man $P$ has $7$ friends,$4$ of them are ladies and $3$ are men. His wife $Q$ also has $7$ friends,$3$ of them are ladies and $4$ are men. Assume $P$ and $Q$ have no common friends. Then the total number of ways in which $P$ and $Q$ together can throw a party inviting $3$ ladies and $3$ men,so that $3$ friends of each of $P$ and $Q$ are in this party,is . . . . . . .