If the letters of the word $ATTEMPT$ are rearranged randomly,what is the probability that all $T$'s occur together?

The number of numbers between $2000$ and $5000$ that can be formed with the digits $0, 1, 2, 3, 4$ (repetition of digits not allowed) and are multiples of $3$ is:

The total number of two-digit numbers $n$ such that $3^{n} + 7^{n}$ is a multiple of $10$ is ..... .

$A$ man has $7$ relatives,$4$ of them are ladies and $3$ are gents; his wife has $7$ other relatives,$3$ of them are ladies and $4$ are gents. The number of ways they can invite $3$ ladies and $3$ gents to a party such that there are $3$ of the man's relatives and $3$ of the wife's relatives,is:

Five balls of different colours are to be placed in three boxes of different sizes. Each box can hold all five balls. In how many ways can we place the balls so that no box remains empty?

All possible $5$-digit numbers,each having $5$ distinct digits,are formed using the digits $\{1, 2, 3, 5, 6, 8\}$. Among them,the number of such numbers which are divisible by $3$ but not by $6$ is: