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 remainder of $n^4-2n^3-n^2+2n-26$ when divided by $24$ is:

Numbers between $1$ and $10,000$ are formed using the digits $2$ and $3$ exactly once and the digit $4$ twice. If the numbers thus formed are arranged in increasing order and $x, y$ represent the ranks of $4324$ and $324$ respectively,then $x-y=$

$A$ group of students comprises $5$ boys and $n$ girls. If the number of ways,in which a team of $3$ students can be randomly selected from this group such that there is at least one boy and at least one girl in each team,is $1750$,then $n$ is equal to

The total number of $7$-digit numbers that can be formed using only the digits $1, 2,$ and $3$ such that the sum of the digits is $10$ is:

If $_nP_r = 30240$ and $\binom{n}{r} = 252$,then $(n, r) = \dots$