If $^{n + 1}{C_3} = 2{\,^n}{C_2},$ then $n =$

Let $A=\left[a_{i j}\right], a_{i j} \in Z \cap[0,4], 1 \leq i, j \leq 2$. The number of matrices $A$ such that the sum of all entries is a prime number $p \in(2,13)$ is $........$.

A country has ten smart cities. The government decides to connect all these cities by road. How many roads the government need to construct so that every city is connected to every other city ?

If $^{2n}{C_2}{:^n}{C_2} = 9:2$ and $^n{C_r} = 10$, then $r = $

Number of integral solutions to the equation $x+y+z=21$, where $x \geq 1, y \geq 3, z \geq 4$, is equal to $..........$.

An engineer is required to visit a factory for exactly four days during the first $15$ days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during 1$-15$ June $2021$ is. . . . . .