A committee of $4$ persons is to be formed from $2$ ladies, $2$ old men and $4$ young men such that it includes at least $1$ lady, at least $1$ old man and at most $2$ young men. Then the total number of ways in which this committee can be formed is

In a shop there are five types of ice-creams available. A child buys six ice-creams.               

Statement $-1 :$ The number of different ways the child can buy the six ice-creams is $^{10}C_5.$ 

Statement $-2 :$ The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging $6 \,A's$ and $4 \,B's$ in a row.

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 $........$.

Determine the number of $5$ card combinations out of a deck of $52$ cards if there is exactly one ace in each combination.

A boy needs to select five courses from $12$ available courses, out of which $5$ courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is

In how many ways can $21$ English and  $19$ Hindi books be placed in a row so that no two Hindi books are together