Find the number of solutions?

Question

Find the number of integral solutions for the equation $x_1+ x_2 + x_3 + x_4 + x_5 + x_6 = 31$

where $x_1 ≥ 1, x_2 ≥ 2, x_3 ≥ 2, x_4 ≥ 4, x_5 ≥ 6, x_6 ≥ 5?$

I have no idea how to proceed here ? I read somewhere that this question can also be done using generating function, an approach using generating function's will be appreciated.

Answer 1

If it is given, $x_i \geq a_i,$ we may say $x_i=y_i+a_i(y\geq0). [i=1,2,...,r]$

So, $\sum x_i=n$ can be transformed to $\sum (y_i+a_i)=n\implies \sum y_i=n- \sum a_i=N.$

Now, you know that the number of solutions to this equation as $y \geq 0$ is $\binom{N+r-1}{r-1}.$

Hope this helps you solve the probelem yourslf as it's always more fun to do so.

Answer 2

Let $y_1 = x_1, y_2 = x_2-1, y_3 = x_3-1, y_4 = x_4-3, y_5 = x_5-5, y_6 = x_6 - 4$. Then $y_1+y_2+y_3+y_4+y_5+y_6 = 17$ and we have all $y_i$ positive integers. The number of solutions, by stars and bars method is $\binom{16}{5}$