Number of ways in which $4$ objects out of $6$ distinct objects can be deranged are

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 $:D_n=n!\left(\sum _{i=0}^n\frac{{\left(-1\right)}^i}{i!}\right)$

Let $A=\left\{1,2,3,4,5,6,7\right\}$. The number of surjective functions defined from $A$ to $A$ such that $f\left(i\right)=i$ for atleast four values of $i$ from $i=1,2,......,7$ , is

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'?

How many numbers of five digits can be made with the digits 1,2,3 each of which can be used at most thrice in a number?

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 

There are $6$ letters and $6$ directed envelopes. Find the number of ways in which all letters are put in the wrong envelopes.

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 each in a box, could be placed such that a ball does not go to a box of its own colour is _____