Five-digit numbers are formed using the digits $1, 2, 3, 5, 7$ with repetitions and are written in descending order with serial numbers. For example,the number $77777$ has serial number $1$. Then the serial number of $35337$ is $.........$.

The total number of four-digit numbers such that each of the first three digits is divisible by the last digit is equal to:

If $a_n = \sum_{r=0}^n \frac{1}{^nC_r}$,then $\sum_{r=0}^n \frac{r}{^nC_r}$ equals

$A$ man has $7$ relatives: $4$ ladies and $3$ gentlemen. His wife also has $7$ relatives: $3$ ladies and $4$ gentlemen. In how many ways can they invite a dinner party of $3$ ladies and $3$ gentlemen,such that there are $3$ of the man's relatives and $3$ of the wife's relatives?

The number of ways of distributing $15$ apples to three persons $A, B, C$ such that $A$ and $C$ each get at least $2$ apples and $B$ gets at most $5$ apples is

Let $A = \{(a, b, c) : a, b, c \text{ are non-negative integers and } a + b + 2c = 22\}$. Then $n(A)$ is equal to: