SOLUTION: Simplify (n+5)!/(n+3)!

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Question 1081665: Simplify (n+5)!/(n+3)!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
By definition,

n! = n*(n-1)*(n-2)*...*3*2*1

Whatever value you pick for n, you count your way down til you get to 1. All of the integers in the countdown are multiplied.

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Using similar logic, the expression (n+5)! can be expanded as such

(n+5)! = (n+5)*(n+4)*(n+3)*(n+2)*(n+1)*...*3*2*1

and that can be turned into

(n+5)! = (n+5)*(n+4)*(n+3)!

notice how the red portions match up. They are equal expressions. In other words,

(n+3)! = (n+3)*(n+2)*(n+1)*...*3*2*1

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Let's use that idea to simplify







%28%28n%2B5%29%21%29%2F%28%28n%2B3%29%21%29=%28n%2B5%29%28n%2B4%29

%28%28n%2B5%29%21%29%2F%28%28n%2B3%29%21%29=n%5E2%2B9n%2B20

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Final Answer: n%5E2%2B9n%2B20