Rank these three quantities from least to greatest.

Question

Consider these three quantities:

Ⅰ.  \frac{49!}{48!}

Ⅱ.  \frac{49!}{(7!)^2}

Ⅲ.  \frac{49!}{(42!)(7!)}

Rank these three quantities from least to greatest.

A. I, II, III
B. I, III, II
C. II, I, III
D. II, III, I
E. III, I, II

jfranciscocuencag · Accepted Answer

Just by canceling 7! we can reach to the answer:

Ⅰ.  \frac{49!}{48!} = \frac{49*48!}{48!} = 49

Ⅱ.  \frac{49!}{(7!)^2}

Ⅲ.  \frac{49!}{(42!)(7!)}

Both Ⅱ and Ⅲ have at least one 7!.

Cancel one of them from each.

Hence,  \frac{49!}{7!}  >  \frac{49!}{42!}

B

GMATinsight · Answer

Ⅰ.  \frac{49!}{48!} = \frac{49*48!}{48!} = 49

Ⅱ.  \frac{49!}{(7!)^2} = 7*8*47*23*3*11*43*42!/7!

Ⅲ.  \frac{49!}{(42!)(7!)} = \frac{49*48*47*46*45*44*43*42!}{42!7!}= 7*8*47*23*3*11*43

From the above expressions it clear that II is largest and I is smallest

Answer: Option B

ScottTargetTestPrep · Answer

Simplifying, we have:

I. 49! / 48! = 49

II. 49! / (7! x 7!)

III. 49! / (42! x 7!)

First off, we see that I is the least of the three. Next, we see that since 7! x 7! Is less than 42! x 7!, II is greater than III. Thus, the order of each Roman Numeral from least to greatest is I, III, II.

Answer: B

Crytiocanalyst · Answer

Ⅰ. 49!/48!= 49!/42!*(43*44*.......48) = 49

Ⅱ. 49!(7!)^2=49!(7!)^2

Ⅲ. 49!/(42!)(7!)

IN comparison of 2 and 3 we get 49!/7! > 49!/42!

Since both have 7! in comparison

least valued is obviously 1 followed by 3 and 2 in the least to the greatest order 
Therefore IMO B

luisdicampo · Answer

Deconstructing the Question  
We compare:
I:  49!/48! 
II:  49!/(7!)^2 
III:  49!/(42!\cdot 7!) 
Rank from least to greatest.

Step-by-step  
Quantity I:
 49!/48! = 49

Compare II vs III:
II =  49!/(7!\cdot 7!) 
III =  49!/(42!\cdot 7!)

Both have  49!/7!  in common.
So we compare the extra denominator factors:  7!  vs  42! .
Since  42! \gg 7! , dividing by  42!  makes the value much smaller:
 III < II

Now compare III vs I:
 III = \frac{49!}{42!\cdot 7!} = \frac{43\cdot44\cdot45\cdot46\cdot47\cdot48\cdot49}{7!} 
This is far greater than 49, so:
 I < III

Therefore:
 I < III < II

Answer: B

Rank these three quantities from least to greatest.

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