Let $A = \{x_1, x_2, x_3, x_4\}$ and $B = \{y_1, y_2, y_3, y_4\}$. $A$ function $f: A \to B$ is defined. The number of one-one functions such that $f(x_i) \neq y_i$ for $i = 1, 2, 3, 4$ is equal to:

There are $4$ distinct colored balls and $4$ boxes of the same colors as the balls. In how many ways can the balls be placed in the boxes such that no ball goes into a box of its own color?

The set $S = \{1, 2, 3, \dots, 12\}$ is to be partitioned into three sets $A, B, C$ of equal size such that $A \cup B \cup C = S$ and $A \cap B = B \cap C = C \cap A = \emptyset$. The number of ways to partition $S$ 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:

In an examination,$5$ students have been allotted their seats as per their roll numbers. The number of ways,in which none of the students sits on the allotted seat,is $..........$.

The number of arrangements of all digits of $12345$ such that at least $3$ digits will not come in their original positions is