The number of ways in which a committee of $6$ members can be formed from $8$ gentlemen and $4$ ladies so that the committee contains at least $3$ ladies is

$n$ balls each of weight $w$. When weighed in pairs,the sum of the weights of all possible pairs is $120$. When they are weighed in triplets,the sum of the weights of all possible triplets is $480$. Then $n$ is:

$A$ set contains $(2n + 1)$ elements. The number of subsets of the set which contain at most $n$ elements 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

$A$ company has $10$ employees. The company has decided to form a team including at least three employees and also excluding at least three employees. Then the number of ways of forming the team is

The least value of $n$ such that ${}^{(n-1)}C_6 + {}^{(n-1)}C_7 < {}^{n}C_8$ is