A student is to answer $10$ out of $13$ questions in an examination such that he must choose at least $4$ from the first five question. The number of choices available to him is
$140$
$196$
$280$
$346$
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 ...... .
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
An engineer is required to visit a factory for exactly four days during the first $15$ days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during 1$-15$ June $2021$ is. . . . . .
The number of four lettered words that can be formed from the letters of word '$MAYANK$' such that both $A$'s come but never together, is equal to
In how many ways can a student choose a programme of $5$ courses if $9$ courses are available and $2$ specific courses are compulsory for every student?