Is this an integer? 12! / 6!, 12! / 8!, or 12! / (7!5!)

how do I decide which one of those is an integer, without taking the time to multiply each and find out?

I) 12!/6!
II)12!/8!
III) 12!/7!5!

Re: Is this an integer???

Here is the way factorials work.
\(\displaystyle \begin{array}{l} 12! = \left( {12} \right)\left( {11} \right)\left( {10} \right) \cdots \left( 2 \right)\left( 1 \right) = 479001600 \\ 12! = \left( {12} \right)\left( {11} \right)\left( {10} \right)\left( 9 \right)\left( 8 \right)\left( 7 \right)\left( {6!} \right) \\ 6! = 720 \\ 7! = 5040 \\ 5! = 120 \\ \end{array}\)

Re: Is this an integer???

I) 12!/6!
II)12!/8!
III) 12!/7!5!

Suppose it is 6!/4! ?
\(\displaystyle \frac{6!}{4!}=\frac{1\cdot2\cdot3\cdot4\cdot5\cdot6}{1\cdot2\cdot3\cdot4}\)
Now, what cancellation can be done? Does it result in an integer?

Suppose it is 7!/9! ?
\(\displaystyle \frac{1\cdot2\cdot3\cdot4\cdot5\cdot6\cdot7}{1\cdot2\cdot3\cdot4\cdot5\cdot6\cdot7\cdot8\cdot9}\)
Now, what cancellation can be done? Does it result in an integer?