I need to that if there is better way to prove the following:

I am trying to prove that
x
r
= k
and k is a multiple of x only when x is prime.

I said that if x is non-prime, then:
Let a = 6
Let r = 4

6!
(6 – 4)! 4!

= 3(2) x 5 x 4!
(2!) 4!
= 15
It is not divisible by 6

My explanation is:
When a is a non-prime number, then a! is [a x (a-1) x (a-2)…x 2 x 1] and a divides out with at least one of the r and this makes it possible for k to be not divisible by x.

and when x is prime:
For example,
Let a = 7
Let r = 4
7!
(7 – 4)! 4!

= 7 x 6 x 5 x 4!
(3!) 4!

= 35
It is divisible by 7

My explanation is:
When a is a prime number, then a! is [a x (a-1) x (a-2)…x 2 x 1] and a does not divide out with any r and this makes it possible for k to be divisible by x.

I am sure there is an elegant wa to prove this; however, i don't exactly know how to do this...can u help plz?? i really appreciate it..thnx

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.