BodhiAI

Nov. 12, 2019

**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 **