$A$ man has $10$ friends. In how many ways can he invite one or more of them to a party?

$X, Y, Z$ are sets of all positive divisors of $10^{60}, 20^{50}$ and $30^{40}$ respectively. Find $n(X \cup Y \cup Z)$.

Let $A_1, A_2, ......, A_{11}$ be players in a team with their $T$-shirts numbered $1, 2, ......, 11$. One hundred gold coins were won by the team. These coins are to be distributed among the players such that each player $A_i$ gets at least $i$ coins,plus an additional coin (i.e.,at least $i+1$ coins). Additionally,the captain (assume $A_1$) and vice-captain (assume $A_2$) must receive at least $5$ and $3$ coins more than their $T$-shirt numbers respectively. In how many different ways can these coins be distributed?

How many numbers greater than a million can be formed with the digits $2, 3, 0, 3, 4, 2, 3$?

The number of ways in which the letters of the word $ARRANGE$ can be arranged such that both $R$ do not come together is

If $\frac{{}^{n + 2}{C_6}}{{}^{n - 2}{P_2}} = 11$,then $n$ satisfies the equation