The digit in the unit place of the number $2009! + 3^{7886}$ is

If the number of seven-digit numbers,such that the sum of their digits is even,is $m \cdot n \cdot 10^{n}$; $m, n \in \{1, 2, 3, \ldots, 9\}$,then $m+n$ is equal to . . . . . . .

The numbers $1, 2, 3, \ldots, n$ are arranged in a random order. The probability that the digits $1, 2, 3, \ldots, k$ appear as a block in that order is:

$A$ man has $7$ relatives: $4$ ladies and $3$ gentlemen. His wife also has $7$ relatives: $3$ ladies and $4$ gentlemen. In how many ways can they invite a dinner party of $3$ ladies and $3$ gentlemen,such that there are $3$ of the man's relatives and $3$ of the wife's relatives?

The number of $4$-letter permutations formed using the English alphabet such that the number of distinct vowels is equal to the number of distinct consonants,when repetition is allowed,is

$9$ balls are to be placed in $9$ boxes. $3$ boxes are so small that they cannot hold $5$ balls. In how many ways can one ball be placed in each box?