In how many ways can a pack of $52$ cards be divided into four equal groups?

There are $4$ balls of different colors and $4$ boxes of the same colors as the balls. In how many ways can $2$ balls be placed in the boxes such that they do not match their respective colors?

Three letters are dictated to three persons and an envelope is addressed to each of them. The letters are inserted into the envelopes at random so that each envelope contains exactly one letter. Find the probability that at least one letter is in its proper envelope.

Six cards and six envelopes are numbered $1, 2, 3, 4, 5, 6$. Cards are to be placed in envelopes such that each envelope contains exactly one card,no card is placed in the envelope bearing the same number,and the card numbered $1$ is always placed in the envelope numbered $2$. The number of ways this can be done is:

The number of ways of dividing $200$ dissimilar things into $10$ groups each containing $20$ elements is

$n$ distinct items $1, 2, 3, \dots, n$ are arranged in $n$ distinct positions $1, 2, 3, \dots, n$. What is the probability that at least three items are in their correct positions?