In how many ways can $3$ letters be posted in $4$ letter boxes,if all the letters are not posted in the same letter box?

How many words can be formed by taking two identical and two distinct letters from the letters of the word $COMBINATION$?

Let $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Let $x$ be the number of $9$-digit numbers formed using the digits of the set $S$ such that only one digit is repeated and it is repeated exactly twice. Let $y$ be the number of $9$-digit numbers formed using the digits of the set $S$ such that only two digits are repeated and each of these is repeated exactly twice. Then,

For integers $n$ and $r$,let $\binom{n}{r} = \begin{cases} ^{n}C_{r}, & \text{if } n \geq r \geq 0 \\ 0, & \text{otherwise} \end{cases}$. The maximum value of $k$ for which the sum $\sum_{i=0}^{k}\binom{10}{i}\binom{15}{k-i} + \sum_{i=0}^{k+1}\binom{12}{i}\binom{13}{k+1-i}$ exists,is equal to ...... .

The word $UNIVERSITY$ is arranged randomly. The probability that both $I$s do not come together is:

The total number of $5$-digit numbers, formed by using the digits $1, 2, 3, 5, 6, 7$ without repetition, which are multiples of $6$, is