How many integer-valued solutions are there?

Question

How many integer-valued solutions are there?

$$ x_1 + x_2 + x_3 + x_4 + x_5 \le 63, \qquad \forall i,\,x_i > 0 $$

My approach:

I think it's "Stars and Bars" Problem. So, when the sum is $63$, we have $63+5-1 \choose 5-1$ = $68 \choose 4$.

And if the sum is $62$, it's $67 \choose 4$.

So the integer-valued solutions are... $$ {68 \choose 4} + {67 \choose 4} + \cdots + {5 \choose 4} + {4 \choose 4}$$ Uhmm am I right...?? I think the answer should be a simple combinatorial function like... ${69\choose 5}$.

Answer 1

Presumably, you mean nonnegative integer-valued solutions.

It can be done in one shot, using a dummy variable $x_6$ to take up the slack.

Explicitly, the number of $5$-tuples $(x_1,...,x_5)$ of nonnegative integers satisfying $$x_1 + \cdots + x_5 \le 63$$ is the same as the number of $6$-tuples $(x_1,...,x_6)$ of nonnegative integers satisfying $$x_1 + \cdots + x_5 + x_6 = 63$$ Finish by using the stars-and-bars formula to count the number of nonnegative integer solutions to the above equation.