Inclusion-Exclusion Principle

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 $n=$?

How many different words can be made out of the letters of INVOLUTE so that each word may contain $3$ vowels and $2$ consonants?

How many different algebraic expressions can be made by combining the letters $a,b,c,d$ and $e$ in this order with the signs $\text{'}+\text{'}$ and $\text{'}-\text{'}$ all the letters taken together?

From $3$ capital letters, $5$ consonants and $4$ vowels, all being different, how many words (i.e., arrangements) can be made each containing $3$ consonants, $2$ vowels and beginning with a capital letter?

In how many ways can $4$ letters be selected out of the letters of the ‘EXPRESSION’?

How many arrangements can be made out of the letters of the word COMMITTEE, taken all at a time, such that the $4$ vowels do not come together?

How many of these words will not contain the two L's together?

In how many ways  can the letter of the word INDEPENDENCE be arranged so that

the words begin with I and end with P?

In how many ways  can the letter of the word INDEPENDENCE be arranged so that

the vowels never occur together ?

In how many ways can the letter of the word INDEPENDENCE be arranged so that the words begin with P?

In how many ways can the letters of the word ARRANGE he arranged so that

neither two A's nor two R's; are together 

The 'cylinder' of a 'Letter-lock' contains $5$ rings. On each ring 6 different letters are engraved. How many unsuccessful attempts may be made to open the lock by a person who does not know the 'key-word'?

There are unlimited number of identical balls of four different colours. How many arrangements of at most $8$ balls in a row can be made by using them?

Puneet writes letters to his five friends and address the corresponding envelopes. If $\lambda$ be the numbers of total ways in which the letters can be placed in the envelopes so that at least three of them are in the wrong envelopes, then the value of $\lambda -100$ 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.

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