How do you simplify #(14!)/(13!)#?

2 Answers
Mar 6, 2018

# "This is the result:" \qquad \qquad \qquad \quad { 14! }/{ 13! } \ = \ 14. #

Explanation:

# "Watch how this goes -- these can be fun ... " #

# \qquad \qquad \qquad \qquad \qquad \ { 14! }/{ 13! } \ = \ { 14 cdot 13 cdot 12 cdot 11 cdot cdots cdot 3 cdot 2 cdot 1 }/{ 13 cdot 12 cdot 11 cdot cdots cdot 3 cdot 2 cdot 1 } #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ { 14 cdot ( 13 cdot 12 cdot 11 cdot cdots cdot 3 cdot 2 cdot 1 ) }/{ ( 13 cdot 12 cdot 11 cdot cdots cdot 3 cdot 2 cdot 1 ) } #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ { 14 cdot ( 13 cdot 12 cdot 11 cdot cdots cdot 3 cdot 2 cdot 1 ) }/{ 1 cdot ( 13 cdot 12 cdot 11 cdot cdots cdot 3 cdot 2 cdot 1 ) } #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ { 14 cdot color{red}cancel{ ( 13 cdot 12 cdot 11 cdot cdots cdot 3 cdot 2 cdot 1 ) } }/{ 1 cdot color{red}cancel{ ( 13 cdot 12 cdot 11 cdot cdots cdot 3 cdot 2 cdot 1 ) } } #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ { 14 }/{ 1 } #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ 14. #

# "Done !!" #

# "So we have our result:" #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ { 14! }/{ 13! } \ = \ 14. #

Jul 8, 2018

#14#

Explanation:

The answer to this question below is a more systematic approach that's great, especially if you're new to factorials, but we have a special case here:

#(14!)/(13!)#

This is in the form

#((a+1)!)/(a!)#, where in our example, #a=13#.

If we were to expand this out, every term would cancel except for the first one. Here's what I mean:

In our example, we essentially have

#(14xx13xx12xx11xx10xx...xx3xx2xx1)/(13xx12xx11xx10xx9xx...xx3xx2xx1)#

Notice, every term on the top and bottom would cancel except for the #14#, because every term the denominator has, so does the numerator.

In general, #((a+1)!)/(a!)# simplifies to #a#, so if you have

#(21!)/(20!)#, this would be #21#.

If we had #(47!)/(46!)#, this would be #47#.

Hope this helps!