# where can i use the  multinomial theorem. can i use it in every type of question or not?expecting a quick reply

Aman Bansal
592 Points
11 years ago

Dear Raman,

## DERANGEMENTS

Any change in the existing order of things is called a derangement.

If ''n'' things are arranged in a row, the number of ways in which they can, be deranged so that none of them occupies its original place is

and it is denoted by D(n).

A question on derangement can be of the following kind:

Illustration:

Supposing 4 letters are placed in 4 different envelopes. In how many ways can be they be taken out from their original envelopes and distributed among the 4 different envelopes so that no letter remains in its original envelope?

Solution:

Using the formula for the number of derangements that are possible out of 4 letters in 4 envelopes, we get the number of ways as :

4!(1 - 1 + 1/2! - 1/3! + 1/4!) = 24(1 - 1 + 1/2 - 1/6 + 1/24) = 9.

## MULTINOMIAL THEOREM

Let x1, x2, ......, xm be integers. Then number of solutions to the equation

x1 + x2 + ... + xm = n                                                                    ... (1)

subject to the conditions a1 < x1 < b1, a2 < x2 < b2, ..., am < xm < bm  ... (2)

is equal to the coefficient of xn in

...(3)

This is because the number of ways in which sum of m integers in (1) subject to given conditions (2) equals n is the same as the number of times xn comes in (3). Using this we get the number of non negative integral solutions of (1) is given byn+m-1Cm-1 and number of positive integral solutions of (1) is given by n-1Cm-1.

Cracking IIT just got more exciting,It s not just all about getting assistance from IITians, alongside Target Achievement and Rewards play an important role. ASKIITIANS has it all for you, wherein you get assistance only from IITians for your preparation and win by answering queries in the discussion forums.

So start the brain storming…. become a leader with Elite Expert League ASKIITIANS

Thanks

Aman Bansal