The number of ways, in which the letters $A, B, C, D, E$ can be placed in the $8$ boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is :

There are $m$ books in black cover and $n$ books in blue cover, and all books are different. The number of ways these $(m+n)$ books can be arranged on a shelf so that all the books in black cover are put side by side 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.

The least value of a natural number $n$ such that $\left(\frac{n-1}{5}\right)+\left(\frac{n-1}{6}\right) < \left(\frac{n}{7}\right)$, where $\left(\frac{n}{r}\right)=\frac{n !}{(n-r) ! r !}, i$

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

If $^{20}{C_{n + 2}}{ = ^n}{C_{16}}$, then the value of $n$ is