# the number of non negative integral solutions of x + y + z ≤ n

## This problem is discussed at

http://www.beatthegmat.com/how-many-non-negative-integral-solutions-are-possible-t34635.html

as you could have discovered with about 5 minutes' work.

## To find the number of non-negative integral solutions of the equation x + y + z ≤ n, we can use the concept of stars and bars or balls and urns.

The stars and bars method is a combinatorial technique that helps count the number of ways to distribute objects into different groups. In this case, we can think of the variables x, y, and z as objects, and the sum x + y + z as the total number of objects we want to distribute.

1. Let's consider the situation when n is greater than or equal to zero.

2. We can start by imagining a row of (n + 1) stars, which represents the value of n. Place these stars in a row.

3. Now, we need to create two dividers (or bars) to distribute the stars into three different groups representing the variables x, y, and z. The position of these dividers divides the sequence of stars into three groups.

4. To calculate the number of solutions, we can count the number of ways to place the dividers among the stars. The number of stars on the left side of the first divider will represent the value of x, the number of stars between the first and second dividers will represent the value of y, and the number of stars on the right side of the second divider will represent the value of z.

5. Since there are two dividers and (n + 1) stars, the total number of positions to place the dividers is (n + 1)C2 or C(n + 1, 2), which is the binomial coefficient of (n + 1) choose 2.

Therefore, the number of non-negative integral solutions of x + y + z ≤ n is given by C(n + 1, 2) or (n + 1)C2.

Note: If n is negative, then the number of non-negative integral solutions of x + y + z ≤ n will be zero, as the inequality cannot be satisfied.