(91!-90!+89!)/89!= : GMAT Problem Solving (PS)
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# (91!-90!+89!)/89!=

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Magoosh GMAT Instructor
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18 Oct 2012, 11:04
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$$\frac{91!-90!+89!}{89!}=$$

A. 2
B. 90
C. 89^2+89
D. 90^2+1
E. 91^1-1

For a full discussion of this and other problems like it, see
http://magoosh.com/gmat/2012/gmat-factorials/

Mike
[Reveal] Spoiler: OA

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Mike McGarry
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Last edited by Bunuel on 18 Oct 2012, 11:08, edited 1 time in total.
Renamed the topic and edited the question.
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18 Oct 2012, 11:11
$$\frac{91!-90!+89!}{89!}=$$

A. 2
B. 90
C. 89^2+89
D. 90^2+1
E. 91^1-1

Factor out 89!: $$\frac{91!-90!+89!}{89!}=\frac{89!(90*91-90+1)}{89!}=90*91-90+1$$.

Now, factor out 90: $$90*91-90+1=90(91-1)+1=90^2+1$$.

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17 Jan 2013, 10:20
$$\frac{91!-90!+89!}{89!}$$ =
(A) 2
(B) 90
(C) 89^2 + 89
(D) 90^2 + 1
(E) 91^2 - 1

For calculation tips involving factorials, as well as a complete solution to this problem, see this post:
http://magoosh.com/gmat/2012/gmat-factorials/

Mike
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Mike McGarry
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17 Jan 2013, 10:40
break up the fraction;

(91!)/(89!) - (90!)/(89!) - (89!)/(89!)
which reduces to (91)(90) - (90) +1 = 90^2 + 1 (D)
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17 Jan 2013, 11:26
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take out the common factor from numerator and denominator which in this case would be 89!

91*90* 89! - 90*89! + 89!

89![ 91*90 - 90 +1]/89!

Which leaves you with

91*90 - 89
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05 Sep 2014, 21:45
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08 Sep 2014, 21:12
$$\frac{91! - 90! + 89!}{89!} = \frac{91!}{89!} - \frac{90!}{89!} + \frac{89!}{89!}$$

$$\frac{91!}{89!}$$ > Units place = 0

$$\frac{90!}{89!}$$ > Units place = 0

$$\frac{89!}{89!} = 1$$

Answer should have 1 in the units place; only option D stands out

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29 Jan 2015, 10:42
Factor out 89!: 91!-90!+89!/89! = 89!(90*91-90+1)/89! = 90*91-90+1.

Now, factor out 90: 90*91-90+1=90(91-1)+1=90^2+1.

Where does the 89! on the top come from?? Is it not true that if we want to get rid of the 89! at the bottom we multiply the top by 89! ? But in this answer we multiply the top by 89! but the 89! remains also at the bottom??
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29 Jan 2015, 11:15
JeroenReunis wrote:
Factor out 89!: 91!-90!+89!/89! = 89!(90*91-90+1)/89! = 90*91-90+1.

Now, factor out 90: 90*91-90+1=90(91-1)+1=90^2+1.

Where does the 89! on the top come from?? Is it not true that if we want to get rid of the 89! at the bottom we multiply the top by 89! ? But in this answer we multiply the top by 89! but the 89! remains also at the bottom??

Dear JeroenReunis,
I'm happy to respond.

First of all, my friend, in your first line of text, you have a mathematical mistake that reflects a misunderstanding of mathematical grouping symbols. See this blog for more information:
https://magoosh.com/gmat/2013/gmat-quan ... g-symbols/

Now, I believe you are misunderstanding the nature of factorials. We did NOT multiply the numerator by 89! in order to cancel it--- you are perfectly correct that this move would have been quite illegal.

Instead, we factored out numbers from the factorial. This blog, the blog from which this question is taken, explains all this in detail:
https://magoosh.com/gmat/2012/gmat-factorials/

Think about what, for example, (91!) means. This is the product of all the positive integers from 91 down to 1. That product would be a very large number, much larger than 10^100 (a googol). We could represent this as

91! = 91*90*89*88*87* ..... *5*4*3*2*1

We have 91 factors all multiplied together. Well, we can group multiplication into any arrangement we like (technically, this is known as the associative property of multiplication). For example,

91! = 91*90*(89*88*87* ..... *5*4*3*2*1)

Well, that set of terms grouped in the parentheses equal 89! Thus,

91! = 91*90*(89!)

Similarly,

90! = 90*(89!)

Thus,

91! + 90! + 89! = 91*90*(89!) + 90*(89!) +(89!) = (91*90 + 90 + 1)*(89!)

That's the precise origin of the (89!) factor in the numerator. We factor it out from the rest of the numerator, and cancel it legitimately with the denominator.

Does all this make sense?
Mike
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29 Jan 2015, 12:00
Hello Mike,

Thank you so much Mike! I really have broken my brains on this one for at least an hour or so..
Now I see my mistake it all makes way more sense.

Re: (91!-90!+89!)/89!=   [#permalink] 29 Jan 2015, 12:00
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