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

A scientific committee is to be formed from $6$ Indians and $8$ foreigners, which includes at least $2$ Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is

Out of $10$ white, $9$ black and $7$ red balls, the number of ways in which selection of one or more balls can be made, is

A group consists of $4$ girls and $7$ boys. In how many ways can a team of $5$ members be selected if the team has at least one boy and one girl ?

If $^{2n}{C_3}:{\,^n}{C_2} = 44:3$, then for which of the following values of $r$, the value of $^n{C_r}$ will be 15

On the occasion of Deepawali festival each student of a class sends greeting cards to the others. If there are $20$ students in the class, then the total number of greeting cards exchanged by the students is