The number of ways in which one or more balls can be selected out of $10$ white,$9$ green,and $7$ blue balls are:

The number of four-letter words that can be formed from the letters of the word '$MAYANK$' such that both '$A$'s are included but never together is equal to:

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:

In a certain test,there are $n$ questions. In the test,$2^{n-i}$ students gave wrong answers to at least $i$ questions,where $i = 1, 2, \dots, n$. If the total number of wrong answers given is $2047$,then $n$ is equal to

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?

$A$ country has $10$ smart cities. The government decides to connect all these cities by road. How many roads does the government need to construct so that every city is connected to every other city?