To prove that the equation x! / ((x - r)! * r!) = k is true only when x is a prime number, we can use the concept of prime factorization.
First, let's assume that x is a non-prime number. We can choose a specific value for x and r to illustrate this. Let's say we set x = 6 and r = 4.
Substituting these values into the equation, we have:
6! / ((6 - 4)! * 4!) = 6! / (2! * 4!)
Expanding the factorials, this becomes:
(6 * 5 * 4 * 3 * 2 * 1) / (2 * 1 * 2 * 1 * 1) = 3 * 5 = 15
We see that 15 is not divisible by 6. This shows that if x is a non-prime number, the resulting value of k may not be divisible by x.
Now, let's consider the case when x is a prime number. We can choose a different value for x and r as an example. Let's say we set x = 7 and r = 4.
Substituting these values into the equation, we have:
7! / ((7 - 4)! * 4!) = 7! / (3! * 4!)
Expanding the factorials, this becomes:
(7 * 6 * 5 * 4 * 3 * 2 * 1) / (6 * 5 * 4 * 3 * 2 * 1) = 7
We see that the resulting value of k is equal to x itself, which is 7 in this case. This shows that if x is a prime number, the resulting value of k will always be divisible by x.
To summarize, we have shown through specific examples that if x is a non-prime number, k may not be divisible by x, while if x is a prime number, k will always be divisible by x. This demonstrates that the equation x! / ((x - r)! * r!) = k is true only when x is a prime number.