Choose the correct number of ways in which $15$ different books can be divided into $5$ heaps of equal number of books.

$A$ person writes letters to $6$ friends and addresses the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that at least two of them are in the wrong envelopes?
Notation : $D_n = n! \left( \sum_{i=0}^n \frac{(-1)^i}{i!} \right)$

There are $4$ envelopes with addresses and $4$ corresponding letters. The probability that no letter goes into its correct envelope is:

Eight persons are to be transported from city $A$ to city $B$ in three cars of different makes. If each car can accommodate at most three persons,then the number of ways,in which they can be transported,is $...........$.

There are $7$ greeting cards, each of a different colour, and $7$ envelopes of the same $7$ colours as the cards. The number of ways in which the cards can be put in envelopes, so that exactly $4$ of the cards go into envelopes of the respective colour, is:

If there are $5$ letters written to $5$ different people and $5$ envelopes addressed to them,then the number of ways in which these letters can be arranged so that no letter goes into its corresponding envelope is