Algebra Ratio of Factorials

Question

Simplify the ratio of factorials.

(n + 7)! / (n + 5 )! =

 

 

 

Answer 1

Marilyn's answer is correct but the explanation is not correct; she shows the set of decreasing factors of (n+7)! as ending with (n+1) and that is incorrect.  There are n+7 factors, and the correct representation of the ending of this has to be 

...(3)(2)(1), etc.

 

All factors of the numerator are cancelled by identical factors of the denominator, except the first two: (n+7)(n+6).

 

 

Answer 2

(n+7)! = (n+7)(n+6)(n+5)(n+4)(n+3)(n+2)(n+1)

 

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

 

since you are dividing the top and bottom, everything on the top gets cancelled by the bottom except (n+7)(n+6).

 

so, (n + 7)! / (n + 5 )! =  (n+7)(n+6) = n²+13n+42.