Inclusion-Exclusion Principle

Suppose four balls labelled $1, 2, 3, 4$ are randomly placed in boxes $B_1, B_2, B_3, B_4$. The probability that exactly one box is empty is

For any two events $A, B$ if $P(A\cup B)=aP(A\cap B)+bP\left(A\right)+cP\left(B\right),$ then $3a+2b+5c=?$

Find the number of ways of selecting a team of $10$ players out of $22$ players if $6$ particular players are always to be included and $4$ particular players are always excluded.

Cotyledons are also called-

The number of different ways in which five alike dashes and eight alike dots can be arranged, using only seven of these ‘dashes’ and ‘dots’ is equal to

If four letters are placed into 4 addressed envelopes at random, the probability that exactly two letters will go wrong is

There are four different boxes A, B, C and D and four different balls $\alpha , \beta ,\gamma$ and $\delta .$ The number of ways of filling the balls in the boxes such that each box contains one balls, ball $\alpha$ can not go to the box A and ball $\beta$ can not go to the box B, is

Let set $A=\{x_1,x_2,x_3,x_4\}$and set  $B=\{y_1, y_2, y_3,y_4\}$. If function is defined from set $A$ to set $B$. Number of one-one function such that $f\left(x_i\right)\ne y_i$ for $i=\mathrm{1,2},\mathrm{3,4}$ is equal to

Number of ways by which $4$ letters can be put in $4$ corresponding envelopes so that all letters go in wrong envelope is

Seven people leave their bags outside a temple and after returning picked one bag each at random. In how many ways at least, one and atmost three of them get their correct bags?

6 letters are to be posted in three letter boxes. The number of ways of posting the letters when no letter box remains empty is

If in a group of five married couples each male shakes hand with exactly one female similarly each female shakes hand with exactly one male, then the probability that no husband shake hands with his own wife, is

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 $4$ of them are in wrong envelopes?

$7$ people leave their bags outside temple and returning after worshiping the deity picked one bag each at random. In how many ways at least one and at most three of them get their correct bags?

A question paper is divided into two parts $A$ and $B$ and each part contain $5$ questions. The number of ways in which a candidate can answer $6$ questions selecting at least two questions from each part is

The number of ways can $10$ letters be placed in $10$ marked envelopes, so that no letter is in the right envelope are

A box contains two white balls, three black balls and four red balls. The number of ways in which three balls can be drawn from the box if at least one black ball is to be included in the draw is

There are four balls of different colours and four boxes of colours same as those of the balls. The number of ways in which the balls, one in each box, could be placed such that a ball does not go to box of its own colour, is

There are $5$ letters and $5$ different envelopes. The number of ways in which all the letters can be put in wrong envelope, is