The number of seven-digit integers with the sum of the digits equal to $10$ and formed by using the digits $1, 2,$ and $3$ only is:

The number of seven-digit integers,with the sum of the digits equal to $10$ and formed by using the digits $1, 2,$ and $3$ only,is

The number of three-digit numbers $\overline{abc}$ such that the arithmetic mean of $b$ and $c$ is equal to the square of their geometric mean is

The number of permutations of the letters $a_1, a_2, a_3, a_4, a_5$ in which the first letter $a_1$ does not occupy the first position and the second letter $a_2$ does not occupy the second position is:

Three schools send $2, 4,$ and $6$ students,respectively,to a summer camp. The $12$ students must be accommodated in $6$ rooms numbered $1, 2, 3, 4, 5, 6$ in such a way that each room has exactly $2$ students and both are from the same school. The number of ways the students can be accommodated in the rooms is

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 $.........$.