Correct Option (D)
The problem requires determining the number of distinct integer triples (x, y, z) that satisfy the equation x + y + z = 6, given that x, y, and z are natural numbers. Natural numbers are defined as positive integers (i.e., x ≥ 1, y ≥ 1, z ≥ 1).
This is a combinatorial problem that can be solved using the Stars and Bars method. For an equation of the form x₁ + x₂ + ... + x_m = n, where each x_i is a natural number, the number of distinct solutions is given by the formula:
Number of solutions = n-1Cm-1
In the given equation:
- n = 6 (the sum)
- m = 3 (the number of variables: x, y, z)
Applying the formula:
Number of solutions = (6-1)C(3-1)
= 5C2
= (5 × 4) / (2 × 1)
= 10
Therefore, there are 10 distinct triples (x, y, z) of natural numbers that satisfy the equation x + y + z = 6.
Incorrect Options:
Options A (4), B (5), and C (9) are incorrect because they do not represent the accurate number of solutions derived from the application of the Stars and Bars formula for natural number partitions. These values may result from an incorrect interpretation of the problem's constraints, such as including zero as a possible value for x, y, or z, or from computational errors. The specification that x, y, and z must be natural numbers (positive integers) is fundamental in determining the correct number of solutions.
