BodhiAI
Nov. 12, 2019

#### Multinomial Theorem:

Multinomial Theorem

If we wish to expand (x+y+z)n, where n is a positive integer and x, y, z are complex numbers, we will follow the way given below:

Step I We will substitute (y+z) = a.

Step II We will expand the Binomial (x+a)n.

Step III Expand a (y+z)r () in each term of the Binomial Expansion (x+a)n.

Hence, In General, if the multinomial expression is of the form then its expansion can be written as

such that, are all +ve integers, and

Problem Solving Trick

Binomial as a special case of Multinomial

If we have only two terms in multinomial expression rather than having m (multiple) value, the expansion in that can will be written as under

where,

which the equivalent form of Binomial Theorem (x+a)n.

Properties of Binomial Coefficients

P-1 If then

either r = s or r + s = n

P-2 Greatest Value of

is greatest if r =

P-3

P-4

P-5

P-6

Reasoning: Since, substituting x= 1, we have

P-7 In the expansion of (1+x)n the sum of even coefficients equals the sum of odd coefficients.

Reasoning: Since, substituting x = -1, we have

P-8

Reasoning: Since, {using P-6}

Also, {using P-7}

Combining, we have

P-9

Reasoning: Since,

Differentiating w.r.t. x, we have

.....(1)

Now, putting x = 1, we have

P-10

Reasoning: Put x = -1 in equation (1) of the last property