$A$ 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 an election,the number of candidates is $1$ greater than the persons to be elected. If a voter can vote in $254$ ways,then the number of candidates is:

There are $5$ types of ice creams available in a shop. $A$ child buys $6$ ice creams.
Statement-$1$: The child can buy $6$ ice creams in $^{10}C_5$ ways.
Statement-$2$: The number of ways the child can buy $6$ ice creams is equal to the number of ways to arrange $6$ '$A$'s and $4$ '$B$'s in a row.

The solution set of ${ }^5 C_{x-1} > 2 \cdot { }^5 C_x$ is

If $\alpha = \binom{m}{2}$,then $\binom{\alpha}{2} = \dots$

There are two urns. Urn $A$ contains $3$ distinct red balls,and Urn $B$ contains $9$ blue balls. If two balls are selected at random from each urn and swapped,in how many ways can this operation be performed?