Principle of Inclusion and Exclusion

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

In a group of $6$ boys and $4$ girls, a team consisting of four children is formed such that the team has atleast one boy. The number of ways of forming a team like this is

The total number of ways in which a student can select a book is $63.$ If he is allowed to select at most $n$ books from a collection of $2n+1$ books, then $n=$?

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 total number of ways in which a student can select a book is $63.$ If he is allowed to select at most $n$ books from a collection of $2n+1$ books, then $n=$?

Determine the number of $5$ cards combinations out of a deck of $52$ cards if at least one of the $5$ cards has to be a king?

A box contains $2$ oranges, $3$ hard-apples and $4$ apples; and the fruits are of different sizes.

In how many ways can the fruits be selected by taking at least one fruit of each kind?

A box contains $2$ oranges, $3$ hard-apples and $4$ apples; and the fruits are of different sizes.
In how many ways can one or more fruits be selected?

Find the total number of selections of at least one black ball from $5$ black balls and $4$ red balls if the balls of the same colour are different.

From $3$ mangoes, $4$ oranges and $2$ apples, how many selections of fruits can be made, taking at least one of each kind? [Assume fruits of the same kind to be a different shapes.]

A man has $5$ oranges and $4$ mangoes. How many different selections having at least one orange are possible?

Out of $4$ officers and $8$ clerks how many selections of $6$ persons can be made so that at least one officer is selected?

If $5$ identical apples, $4$ identical oranges and $3$ identical bananas in a fruit basket. The number of ways a person can select at least $2$ apples, $2$ oranges and $1$ banana is

Let $E$ denote the set of letters of the English alphabet, $V=\left\{a,e,i,o,u\right\}$ and $C$ be the complement of $V$ in $E$. Then, the number of four-letter words (where repetitions of letters are allowed) having at least one letter from $V$ and at least one letter from $C$ is

Find the number of ways of selecting a cricket team of $11$ players from $7$ batsman and $6$ bowlers such that there will be atleast $5$ bowlers in the team.

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

In a class, there are total $7$ boys and $5$ girls. The number of different teams of $2$ girls and $3$ boys that can be formed from this class, if there are two specific boys $A$ and $B,$ who refused to be the members of the same team, is:

The number of ways in which $3$ book can be placed in $4$ racks with at most one book in each rack is:

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.

A shopkeeper places $41$ different toys in front of you out of which $20$ toys are to be purchased. Suppose $\mathrm{m}$ is the number of ways in which $20$ toys can be purchased without any restriction and $\mathrm{n}$ is the number of ways in which a particular toy is to be always included in each selection of $20$ toys, then $(\mathrm{m}-\mathrm{n})$ can be expressed as