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

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, ......, x_m be integers. Then number of solutions to the equation

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

subject to the conditions a₁ < x₁ < b₁, a₂ < x₂ < b₂, ..., a_m < x_m < b_m  ... (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 by^n+m-1C_m-1 and number of positive integral solutions of (1) is given by ^n-1C_m-1.

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Multinomial theorem is used when you need to do selection from alike objects.