 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