Since the question is about multinomial theorem , awareness about binomial theorem is assumed. Binomial theorem is about the expansion of (x+y)ⁿ. In this expansion the term containing x^n-ry^r is given by ⁿC_r x^n-ry^r . The value of the coefficient ⁿC_r is written as follows:

. This implies the total number of ways of selecting r objects of one type and n-r objects of another type out of n objects. This is the coefficient of the term containing x^n-ry^r

Similar principle applies when a multinomial is expanded. Thus Multinomial Theorem is about expanding the positive integral power of a sum of terms. A term in the expansion of (x₁+x₂+x₃+…x_m)ⁿ containing (x₁)^k1 (x₂)^k2 ..(x_m)^km (( such that k₁ +k₂ +…k_m = n) , will have a coefficient which would imply the total number of ways of selecting k₁ objects of the type x₁ , k₂ objects of the type x₂ …etc out of n objects ( such that k₁ +k₂ +…k_m = n) . The value of this coefficient is written as follows

. The term containing (x₁)^k1 (x₂)^k2 ..(x_m)^km will be

(x₁)^k1 (x₂)^k2 ..(x_m)^km.

As an example we can write the coefficient of term containing a³ (implying a³ b⁰ c⁰) in the expansion of (a+b+c)³ as

. The coefficient of the term containing a²b¹c⁰ will be . Similarly, the coefficient of the term containing abc will be

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