Hey! First of all, I'm terribly sorry if this is in the wrong category! I'm not sure where this fits in.
I'd like some help with understanding binominal coefficients.
So, if I've understood it correctly, the general formula for any binominal coefficient is
(n choose k) = n!/k!(n-k)!
But I'm having trouble understanding the logic behind the formula. I understand that putting your desired numbers k and n in said formula will give the correct answer, but why is that? How can you see from looking at the formula that it'll give you the amount of different k's in n? Additionally, as I've understood it, any binominal coefficients where the second integer is 2 will always give
(n choose 2) = n(n-1)/2
Is this just because the general formula always gives that answer, or is there another reason? Why just n(n-1)? I was originally thinking it had something to do with the fact that n! = n*(n-1)!, but that wouldn't seem to be the case, given that, in the formula, n(n-1) isn't factorial. Also, why divided by 2? Is that also just because applying the general formula will always leave 2 as the denominator, in this case, or is there another way to realize that?
Sorry if I'm unclear, and again, if I'm in the wrong category.
Thanks in advance!
I'd like some help with understanding binominal coefficients.
So, if I've understood it correctly, the general formula for any binominal coefficient is
(n choose k) = n!/k!(n-k)!
But I'm having trouble understanding the logic behind the formula. I understand that putting your desired numbers k and n in said formula will give the correct answer, but why is that? How can you see from looking at the formula that it'll give you the amount of different k's in n? Additionally, as I've understood it, any binominal coefficients where the second integer is 2 will always give
(n choose 2) = n(n-1)/2
Is this just because the general formula always gives that answer, or is there another reason? Why just n(n-1)? I was originally thinking it had something to do with the fact that n! = n*(n-1)!, but that wouldn't seem to be the case, given that, in the formula, n(n-1) isn't factorial. Also, why divided by 2? Is that also just because applying the general formula will always leave 2 as the denominator, in this case, or is there another way to realize that?
Sorry if I'm unclear, and again, if I'm in the wrong category.
Thanks in advance!
