For a natural number $n$,the inequality $2^n(n - 1)! < n^n$ holds true if:

Five boys and five girls are writing an examination in a hall in which $5$ benches are arranged in a row and only two students are to be seated on every bench at either of its ends. If the seating arrangement is to be such that no two boys or no two girls sit together as neighbours (a student should not have a student of same gender either on left or right),then the total number of such arrangements is

The number of $5$-digit numbers that can be formed using the digits $1, 2, 3, 4, 5, 6$ such that the number must include both $1$ and $2$ is:

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:

The value of ${}^6P_4 + 4 \cdot {}^6P_3$ is $.......$ .

$A$ certain question paper contains three parts $A, B, C$ with four questions in part $A$,five questions in part $B$,and six questions in part $C$. $A$ student is required to answer seven questions,choosing at least two questions from each part. The total number of different ways a student can choose his seven questions for answering is: