If $n$ and $r$ are two positive integers such that $n \ge r,$ then $^nC_{r-1} + ^nC_r = $

The number of ways in which $3$ children can distribute $10$ tickets out of $15$ consecutively numbered tickets among themselves such that they get consecutive blocks of $5, 3$ and $2$ tickets is:

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

$A$ customer forgets a $4$-digit code for an Automatic Teller Machine ($A$.$T$.$M$.) in a bank. However,he remembers that this code consists of digits $3, 5, 6$ and $9$. Find the largest possible number of trials necessary to obtain the correct code.

In how many ways can $5$ prizes be distributed among four students when every student can take one or more prizes?

In how many different ways can the letters of the word '$OPTICAL$' be arranged so that the vowels always come together?