Question

Find the number of non-negative integral solutions of x + y + z + w = 20.

Answer 1

Any one of the four variables can take values from zero to 20 and hence we construct a polynomial in a variable (say x) with x raised to different powers which would constitute the values that any one variable can take when the equation is solved in the manner indicated. We, thus, consider the product expression

(1 + x + x² + ... + x²⁰)(1 + x + ... + x²⁰)

(1 + x + ....+ x²⁰)(1 + x + ...+ x²⁰)

There are four factors since there are four variables. If we take x⁴ in the first factor, x⁵ in the second, x⁸ in the third, then we take the term x3 in the fourth so that the sum of the powers (4 + 5 + 8 + 3 = 20) is 20. It is then we say that there is a solution corresponding to x = 4, y = 5, z = 8, w = 3. Hence, the number of solutions in the manner required is

Coefficient of x²⁰ in (1 +  x + ....+ x²⁰)⁴

= Coefficient of x²⁰ in  (1 - x²¹/1 - x)⁴

= Coefficient of x²⁰ in (1 - x²¹)⁴(1 - x)^-4

= Coefficient of x²⁰ in (1 - x)^-4

= ²³C₃

[Note: In (1 - x)^-4 coefficient of xⁿ is (n + 3)C₃]