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

Words with or without meaning are to be formed using all the letters of the word $EXAMINATION$. The probability that the letter $M$ appears at the fourth position in any such word is:

If $4$-digit numbers greater than $5,000$ are randomly formed from the digits $0, 1, 3, 5,$ and $7$,what is the probability of forming a number divisible by $5$ when the repetition of digits is not allowed?

$(2n + 1) (2n + 3) (2n + 5) \dots (4n - 1)$ is equal to :

There are $(n + 1)$ white balls and $(n + 1)$ black balls. Each ball is numbered from $1$ to $(n + 1)$. In how many ways can these balls be arranged in a row such that no two balls of the same color are adjacent?

There are $10$ pairs of shoes in a cupboard from which $4$ shoes are picked at random. The probability that there is at least one pair is: