Multinomial expansion:

In the expansion of (x₁+x₂ + . . .  + x_n)^m where m, n Î N  and x₁, x₂ , . . ., x_n are independent variables, we may get

• Total terms in the expansion = ^m+n-1C_n-1
• Coefficient of   (where r₁ + r₂ +...+ r_n = m, r_i ÎN È {0} is   .
• Sum of all the coefficient is calculated by putting all the variables x_i same to 1 and it is similar to n^m.

Illustration: If x₁ + x₂ + x₃ + x₄ + x₅ = 20  and x₁ + x₂ = 5 , (x₁ ,x₂ , x₃ ,x₄ , x₅ ³ 0) then calculate the number of non negative integral solutions of above equation.

Solution:                   x₁ + x₂ + x₃ + x₄ + x₅ = 20 , x₁ + x₂ = 5                    ... (1)

                                    => x₃ + x₄ + x₅ = 15                                                    ... (2)

                                    Number of solutions

=>   Coefficient of x⁵ in (1) ´ coefficient of x¹⁵ in (2)

=>    Coefficient of ´ Coefficient of  

=>   Coefficient of x⁵ in (1 - x)^-2 ´ Coefficient of x¹⁵

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