A committee of $7$ has to be formed from $9$ boys and $4$ girls. In how many ways can this be done when the committee consists of: 

exactly $3$ girls $?$

If $n \geq 2$ is a positive integer, then the sum of the series ${ }^{ n +1} C _{2}+2\left({ }^{2} C _{2}+{ }^{3} C _{2}+{ }^{4} C _{2}+\ldots+{ }^{ n } C _{2}\right)$ is …… .

At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are $10$ candidates and $4$ are of be selected, if a voter votes for at least one candidate, then the number of ways in which he can vote is

If $x,\;y$ and $r$ are positive integers, then $^x{C_r}{ + ^x}{C_{r – 1}}^y{C_1}{ + ^x}{C_{r – 2}}^y{C_2} + …….{ + ^y}{C_r} = $

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.