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# For positive integers n and m, is m!<3^n ?

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Math Expert
Joined: 02 Sep 2009
Posts: 56304
For positive integers n and m, is m!<3^n ?  [#permalink]

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26 Sep 2016, 03:33
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Difficulty:

65% (hard)

Question Stats:

53% (02:12) correct 47% (01:42) wrong based on 151 sessions

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For positive integers n and m, is m! < 3^n ?

(1) n = m

(2) n > 3

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Re: For positive integers n and m, is m!<3^n ?  [#permalink]

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26 Sep 2016, 05:46
1
Bunuel wrote:
For positive integers n and m, is m! < 3^n ?

(1) n = m

(2) n > 3

Stat 1: n = m

m! < 3^n ...this statement will work till 5 = n = m

120 < 3*3*3*3*3

Now if m = 7 and n =7 .

5040 > 2187....so insufficient.

Stat 2: we don't have relationship between n and m ...insufficient.

Both : if we take n > 3...i.e. 4 to 6...it is fine but if it is n = 7...Insufficient.

IMO option E.
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For positive integers n and m, is m!<3^n ?  [#permalink]

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25 Feb 2018, 08:43
2
Bunuel wrote:
For positive integers n and m, is m! < 3^n ?

(1) n = m

(2) n > 3

Nothing much from $$m! < 3^n$$..

let's see statements

1) $$n=m$$..
$$n=m=1.. 1!<3^1$$ so $$1< 3$$ ..yes
each term thereafter the left term m! will increase 4 times then 5 times and even 1000times when it is 1000! WHEREAS right will increase ONLY 3 times with each increase
So at some place Left side will become GREATER than Right side
BOTH $$m! < 3^n$$ and $$m! > 3^n$$ possible
Insuff

2) $$n>3..$$
insufficient

combined..
lets check for $$m=n=4$$..
$$4!<3^4...24<81...$$yes
But at higher values we know that $$m!>3^n$$
insuff

E
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For positive integers n and m, is m!<3^n ?  [#permalink]

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26 Feb 2018, 09:35
Bunuel wrote:
For positive integers n and m, is m! < 3^n ?

(1) n = m

(2) n > 3

Is $$m!<3^n=>\frac{m!}{3^n}<1$$ -------(1)

now $$m!=m(m-1)(m-2)(m-3)....3*2*1$$ and $$3^n=3*3*3$$..... to n terms

so $$\frac{m!}{3^n}=\frac{m(m-1)(m-2)(m-3)....3*2*1}{3*3*3*3.....n terms}$$

Notice that in the above fraction only numerators 2 & 1 are less than 3 and rest all numerators will be either greater than 3 or equal to 3. So if in the numerator we have a number, such as 9, that can take care of the two 3's in the denominator then equation (1) will be definitely greater than 1

Statement 1: $$n=m=4$$, then equation

then $$\frac{m!}{3^n}=\frac{4*3*2*1}{3*3*3*3}$$, clearly $$\frac{4}{3}$$ is marginally greater than $$1$$, $$\frac{3}{3}=1$$ but $$\frac{2}{3}$$ & $$\frac{1}{3}$$ are less than $$1$$. Hence this fraction will be definitely less than $$1$$. So we have a Yes for equation (1). but if $$n=m=10$$

then $$\frac{m!}{3^n}=\frac{10*9*8*7*6*5*4*3*2*1}{3*3*3*3*3*3*3*3*3*3}$$, clearly $$10$$ & $$9$$ will take care of 3's corresponding to numerators $$2$$ & $$1$$. Hence this number will definitely be greater than $$1$$. so we have a No for equation (1). Insufficient

Statement 2: nothing mentioned about $$m$$. Insufficient

Combining (1) & (2): As we can see from statement 1, we will have both the scenarios, hence Insufficient

Option E
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Re: For positive integers n and m, is m!<3^n ?  [#permalink]

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26 Feb 2018, 11:39
Can anyone help me where everyone seeing options???
I can't see them with the question.
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Akanksha Sharma
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Re: For positive integers n and m, is m!<3^n ?  [#permalink]

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26 Feb 2018, 11:41
SharmaAkanksha wrote:
Can anyone help me where everyone seeing options???
I can't see them with the question.

Hi,

In the DS question options are standard. Hence they are not represented again

Posted from my mobile device
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Posts: 56304
Re: For positive integers n and m, is m!<3^n ?  [#permalink]

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26 Feb 2018, 11:45
SharmaAkanksha wrote:
Can anyone help me where everyone seeing options???
I can't see them with the question.

This is a data sufficiency question. Options for DS questions are always the same.

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether—

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

I suggest you to go through the following posts:
ALL YOU NEED FOR QUANT.

Hope this helps.
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Re: For positive integers n and m, is m!<3^n ?  [#permalink]

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22 Aug 2018, 04:10
SharmaAkanksha wrote:
Can anyone help me where everyone seeing options???
I can't see them with the question.

The most common strategy is answers will be either from AD or BCE.
These are DS questions. And are about half of the quant questions (15 out of 31) will be of this type.
Happy to help.
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Re: For positive integers n and m, is m!<3^n ?   [#permalink] 22 Aug 2018, 04:10
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