How do you simplify this Factorial?

Question

Simplify
(2k-2)! / 2k!

Can you show me step by step?

Answer 1

since n! = n x (n-1) x (n-2) x.....
2k! = 2k x (2k-1) x (2k-2)!
so putting this as the denominator
(2k-2)!/{2k x (2k-1) x (2k-2)!
cancel (2k-2)! and you are left with
1/{2k x (2k-1)}
1/2k(2k-1) or 1/4k^2 -2k

Answer 2

The numerator is 1 * 2 * 3 * ... * (2k-2).

The denominator is 1 * 2 * 3 * ... * (2k-2) * (2k-1) * 2k.

Everything in the numerator cancels out, leaving you with (2k-1) * 2k in the denominator. The answer is 1/(2k-1)2k = 1/(4k² - 2k).

Answer 3

Ok, i havent done factorials in a while but ill take a crack at it.

(2k-2)! / 2k!

=

2k!-2! / 2k!

therefore the 2k! in the numerator and the 2k! in the denominator would cancel eachother out and leave you with -2! which is = 2.

I believe thats right, makes sense to me.

Answer 4

expand the denominator 2k! = (2k-2)! * (2k-1) * 2k

(2k-1)! gets cancelled out fron numerator and denominator

so the answer is 1 / ( (2k-1) * 2k )

Answer 5

n! = 1 x 2 x 3 x 4.... x (n-2) x (n-1) x n

So:

[1 x 2 x....x (2k-2)] / [ 1 x 2 x 3 x...x (2k-2) x (2k-1) x 2k] =

1/ [ (2k-1) x 2k]

and that's that.

Answer 6

just apply the definition of the factorial symbol. if you can't see the answer, start by looking at some specific examples. for instance, when k = 2, the expression simplifies to 1/(4*3). What about when k = 3 or 4? the general expression should then be clear to you.

alice j is incorrect; there is no reason why the factorial symbol should distribute over subtraction; in fact, it would be difficult to find an example where this is true.

Answer 7

2k-2
____ = -2
2k
2k on top with 2k in buttom will simplify
two equality amount in fraction means you can erase them.

Answer 8

(2k - 2)! / [(2k - 2)! * (2k - 1) * (2k)] =
1 / (2k - 1)(2k) =
1 / 2k^2 - 2k

Answer 9

(2k - 2)!
--------------------
2k (2k - 1) (2k - 2)!

1
---------------
2k (2k - 1)