If all the letters of the word $'GANGARAM'$ are arranged,then the number of words in which exactly two vowels are together but no two $'G'$ occur together is-

The number of all four-digit numbers which do not have four distinct digits is

Let $m$ and $n$ $(m < n)$ be two $2$-digit numbers. Then the total number of pairs $(m, n)$ such that $\operatorname{gcd}(m, n) = 6$ is . . . . . . .

The sum of all possible numbers that can be formed by using the digits $2, 3, 5, 7$ without repetition of digits 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

Let $n \geq 3$ be an integer. For a permutation $\sigma = (a_1, a_2, \ldots, a_n)$ of $(1, 2, \ldots, n)$,we define $f_\sigma(x) = a_n x^{n-1} + a_{n-1} x^{n-2} + \ldots + a_2 x + a_1$. Let $S_\sigma$ be the sum of the roots of $f_\sigma(x) = 0$ and let $S$ denote the sum over all permutations $\sigma$ of $(1, 2, \ldots, n)$ of the values $S_\sigma$. Then,