If $10 \cdot ^nC_2 = 3 \cdot ^{n+1}C_3$,then the value of $n$ is

There are $5$ apples,$4$ mangoes,$3$ oranges,and $1$ each of $2$ other varieties of fruits. The number of ways of selecting at least one fruit of each kind is

There are $4$ oranges,$5$ apples,and $7$ mangoes in a fruit basket. The number of ways of selecting at least one fruit from among the fruits in the basket is

If $^{2n}C_2 : ^nC_2 = 9:2$ and $^nC_r = 10$,then $r = $

If there are $6$ alike fruits,$7$ alike vegetables,and $8$ alike biscuits,then the number of ways of selecting any number of things out of them such that at least one from each category is selected,is

The number of ordered pairs $(r, k)$ for which $6 \cdot ^{35}C_{r} = (k^{2} - 3) \cdot ^{36}C_{r+1}$,where $k$ is an integer,is