$^{n-1}C_3 + ^{n-1}C_4 > ^nC_3$,then the value of $n$ is

An engineer is required to visit a factory for exactly $4$ days during the first $15$ days of every month,and it is mandatory that no two visits take place on consecutive days. 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 ways in which $21$ identical apples can be distributed among three children such that each child gets at least $2$ apples,is

In how many ways can $16$ rupees be divided among $4$ persons such that no person receives less than $3$ rupees?

$A$ lady gives a dinner party for $6$ guests. The number of ways in which they may be selected from among $10$ friends,if two of the friends will not attend the party together,is:

$A$ student is allowed to select at most $n$ books from a collection of $(2n + 1)$ books. If the total number of ways in which he can select at least one book is $63$,then the value of $n$ is