What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these are two red cards and two black cards?

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

In how many ways can a team of $10$ players be selected from $22$ players if $6$ particular players are always to be included and $4$ particular players are always excluded?

The number of ordered triplets of positive integers which are solutions of the equation $x + y + z = 100$ is

Consider three boxes,each containing $10$ balls labelled $1, 2, \dots, 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $n_i$ the label of the ball drawn from the $i^{th}$ box,$(i = 1, 2, 3)$. Then,the number of ways in which the balls can be chosen such that $n_1 < n_2 < n_3$ is:

The number of ways of forming a committee of $6$ members out of $5$ Indians,$5$ Americans,and $5$ Australians such that there will be at least one member from each country in the committee is: