binaras wrote:Hi
need some help with the following
Question
Which of the following is an integer
1. 12!/6!
2. 12!/8!
3. 12!/7!5!
esp with the 3.12!/7!5!
Thanks
Solution:
Before actually solving this problem, let's review how factorials can be expanded and expressed. As as example, we can use 5!.
5! could be expressed as:
5!
5 x 4!
5 x 4 x 3!
5 x 4 x 3 x 2!
5 x 4 x 3 x 2 x 1!
Understanding how this factorial expansion works will help us work our way through each answer choice, especially answer choices 1 and 2.
1. 12!/6!
Since we know that factorials can be expanded, we now know that:
12! = 12 x 11 x 10 x 9 x 8 x 7 x 6!
Plugging this in for answer choice 1, we have:
(12 x 11 x 10 x 9 x 8 x 7 x 6!)/6! = 12 x 11 x 10 x 9 x 8 x 7, which is an integer.
2. 12!/8!
Once again, since we know that factorials can be expanded, we now know that:
12! = 12 x 11 x 10 x 9 x 8!
Plugging this in for answer choice 2, we have:
(12 x 11 x 10 x 9 x 8!)/8! = 12 x 11 x 10 x 9, which is an integer.
3. 12!/(7!5!)
Once again, since we know that factorials can be expanded, we now know that:
12! = 12 x 11 x 10 x 9 x 8 x 7!
Plugging this in for answer choice 3 gives us:
(12 x 11 x 10 x 9 x 8 x 7!)/(7!5!)
(12 x 11 x 10 x 9 x 8)/(5 x 4 x 3 x 2 x 1)
(12 x 11 x 10 x 9 x 8)/(12 x 10 x 1)
11 x 9 x 8, which
is an integer.
We see that choices 1, 2, and 3 are all integers.
