The equation x + y + z = 16 represents the sum of three non-negative integers, x, y, and z, that add up to 16. We need to find the number of integral solutions for this equation where all x, y, and z are less than 8.
To solve this problem, we can use a technique called stars and bars, or balls and urns.
Imagine having 16 identical balls and 2 dividers (|) to separate them into 3 groups. Each group represents the value of x, y, and z. For example, if we have 4 balls before the first divider, 7 balls between the first and second dividers, and 5 balls after the second divider, it represents x = 4, y = 7, and z = 5.
We can represent this visually as:
**** | ******* | *****
To find the number of solutions, we need to find the number of ways we can arrange these balls and dividers. We can think of this as a permutation problem. In this case, we have 16 balls and 2 dividers, so we have a total of 18 objects to arrange.
Using the formula for permutations with repetition, we can calculate the number of ways to arrange these objects:
N = (n + r - 1) C (r)
where N is the number of ways to arrange the objects, n is the total number of objects to arrange (16 balls + 2 dividers = 18), and r is the number of identical objects to arrange (2 dividers).
Applying this formula, we get:
N = (18 C 2) = 153
Therefore, there are 153 non-negative integral solutions for the equation x + y + z = 16 where all x, y, and z are less than 8