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If n > 4, what is the value of the integer n ?

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gmatt1476
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If n > 4, what is the value of the integer n ? [#permalink] New post   21 Sep 2019, 16:10
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If n > 4, what is the value of the integer n ?


(1) \(\frac{n!}{(n - 3)!} = \frac{3!n!}{4!(n - 4)!}\)

(2) \(\frac{n!}{3!(n - 3)!} + \frac{n!}{4!(n - 4)!} = \frac{(n + 1)!}{4!(n - 3)!}\)




DS47661.01
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chetan2u
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If n > 4, what is the value of the integer n ? [#permalink] New post   19 Oct 2019, 01:05
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If n > 4, what is the value of the integer n ?


(1) \(\frac{n!}{(n - 3)!} = \frac{3!n!}{4!(n - 4)!}\)
Two ways..
(a) Combination formula
\(\frac{n!}{(n - 3)!} = \frac{3!n!}{4!(n - 4)!}\).....\(\frac{n!}{3!(n - 3)!} = \frac{n!}{4!(n - 4)!}.....nC3=nC4\)
Thus \(n=3+4=7\)..SUFF
(b) Arithmetic way
\(\frac{n!}{(n - 3)!} = \frac{3!n!}{4!(n - 4)!}\).......\(\frac{n(n-1)(n-2)(n-3)!}{(n - 3)!} = \frac{3!n(n-1)(n-2)(n-3)(n-4)!}{4!(n - 4)!}\)....\(4!*n(n-1)(n-2)=3!*(n)(n-1)(n-2)(n-3).....4=n-3...n=7..SUFF\)



(2) \(\frac{n!}{3!(n - 3)!} + \frac{n!}{4!(n - 4)!} = \frac{(n + 1)!}{4!(n - 3)!}\)
Two ways..
(a) Combination formula
\(\frac{n!}{3!(n - 3)!} + \frac{n!}{4!(n - 4)!} = \frac{(n + 1)!}{4!(n - 3)!}\).....\(.....nC3+nC4=(n+1)C4\)
Thus n could be anything as it is true for all.
(b) Arithmetic way
\(\frac{n!}{3!(n - 3)!} + \frac{n!}{4!(n - 4)!} = \frac{(n + 1)!}{4!(n - 3)!}\) ......
Take out n!/3!(n-3)!
\(\frac{n!}{3!(n-3)!}(1+\frac{n-3}{4} )=\frac{n!}{3!(n-3)!}(\frac{n+1}{4} ).....\frac{n+1}{4}=\frac{n+1}{4}\)....Always true..
This also tells us why nC3+nC4=(n+1)C4

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Re: If n > 4, what is the value of the integer n ? [#permalink] New post   21 Sep 2019, 16:22
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for statement (1), it can be simplified to:
\(n(n-1)(n-2) = \frac{n(n-1)(n-2)(n-3)}{4}\)
\(4 = n-3\)
\(n = 7\) --> sufficient

for statement (2), upon simplification it will end with \(n = n\), which means no definite value --> insufficient

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If n > 4, what is the value of the integer n ? [#permalink] New post   Updated on: 19 Oct 2019, 01:20
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Statement 1-
nC3=nC4
n=7

sufficient

Statement 2-

nC3+nC4= (n+1)C4
This is true for all n(>4).

{Pascal Rule- nC(k-1) + nCk= (n+1)Ck

Insufficient


gmatt1476 wrote:
If n > 4, what is the value of the integer n ?


(1) \(\frac{n!}{(n - 3)!} = \frac{3!n!}{4!(n - 4)!}\)

(2) \(\frac{n!}{3!(n - 3)!} + \frac{n!}{4!(n - 4)!} = \frac{(n + 1)!}{4!(n - 3)!}\)




DS47661.01

Originally posted by nick1816 on 18 Oct 2019, 16:30.
Last edited by nick1816 on 19 Oct 2019, 01:20, edited 1 time in total.
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Re: If n > 4, what is the value of the integer n ? [#permalink] New post   19 Oct 2019, 00:41
nick - Can you explain why statement B is always true ?

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Re: If n > 4, what is the value of the integer n ? [#permalink] New post   19 Oct 2019, 00:55
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You can select k items from n+1 elements in 2 ways.

Either directly select k items from n+1 elements

or

[you can select k items from n elements] + [select k-1 items from those n element and add (n+1)th item]


Dug50 wrote:
nick - Can you explain why statement B is always true ?

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If n > 4, what is the value of the integer n ? [#permalink] New post   19 Oct 2019, 00:59
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gmatt1476 wrote:
If n > 4, what is the value of the integer n ?


(1) \(\frac{n!}{(n - 3)!} = \frac{3!n!}{4!(n - 4)!}\)

(2) \(\frac{n!}{3!(n - 3)!} + \frac{n!}{4!(n - 4)!} = \frac{(n + 1)!}{4!(n - 3)!}\)




DS47661.01


If n > 4, what is the value of the integer n ?


(1) \(\frac{n!}{(n - 3)!} = \frac{3!n!}{4!(n - 4)!}\)
1/(n-3) = 3!/4! = 1/4
n-3 =4
n=7
SUFFICIENT

(2) \(\frac{n!}{3!(n - 3)!} + \frac{n!}{4!(n - 4)!} = \frac{(n + 1)!}{4!(n - 3)!}\)
n!/4!(n-3)! [4 + n-3] = n!/4!(n-3)! (n+1) = (n+1)!/4!(n-3)!
Always true for all values of n except n=3
NOT SUFFICIENT

IMO A
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Re: If n > 4, what is the value of the integer n ? [#permalink] New post   18 Aug 2020, 05:51
I feel the answer should be D. As on solving 2nd condition,we get certain value of x as 5.

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Re: If n > 4, what is the value of the integer n ? [#permalink] New post   02 Nov 2020, 21:56
Even during the exam if the combination formula does not strike, then also its fine. Most of the stuff cancels out easily. Just be aware of the trap for statement 2.
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Re: If n > 4, what is the value of the integer n ? [#permalink] New post   30 Nov 2020, 06:34
Can some one please explain the simplification proces of the statements step by step? Could you please do not skip any mini step in the middle as well.

Thanks in advance!
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Re: If n > 4, what is the value of the integer n ? [#permalink] New post   03 Feb 2021, 03:03
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Don’t get flummoxed by complicated equations and factorials. Try to break down factorials to cancel a few of them from both sides.

Check the attachment. Answer is A.

Hope this helps. :)
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DS Question.jpg [ 105.19 KiB | Viewed 15548 times ]

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Re: If n > 4, what is the value of the integer n ? [#permalink] New post   05 Feb 2022, 13:21
can someone please explain scenario 2 using the combinations formula? i'm not sure i'm understanding why n could be anything

chetan2u wrote:
If n > 4, what is the value of the integer n ?


(1) \(\frac{n!}{(n - 3)!} = \frac{3!n!}{4!(n - 4)!}\)
Two ways..
(a) Combination formula
\(\frac{n!}{(n - 3)!} = \frac{3!n!}{4!(n - 4)!}\).....\(\frac{n!}{3!(n - 3)!} = \frac{n!}{4!(n - 4)!}.....nC3=nC4\)
Thus \(n=3+4=7\)..SUFF
(b) Arithmetic way
\(\frac{n!}{(n - 3)!} = \frac{3!n!}{4!(n - 4)!}\).......\(\frac{n(n-1)(n-2)(n-3)!}{(n - 3)!} = \frac{3!n(n-1)(n-2)(n-3)(n-4)!}{4!(n - 4)!}\)....\(4!*n(n-1)(n-2)=3!*(n)(n-1)(n-2)(n-3).....4=n-3...n=7..SUFF\)



(2) \(\frac{n!}{3!(n - 3)!} + \frac{n!}{4!(n - 4)!} = \frac{(n + 1)!}{4!(n - 3)!}\)
Two ways..
(a) Combination formula
\(\frac{n!}{3!(n - 3)!} + \frac{n!}{4!(n - 4)!} = \frac{(n + 1)!}{4!(n - 3)!}\).....\(.....nC3+nC4=(n+1)C4\)
Thus n could be anything as it is true for all..SUFF
(b) Arithmetic way
\(\frac{n!}{3!(n - 3)!} + \frac{n!}{4!(n - 4)!} = \frac{(n + 1)!}{4!(n - 3)!}\) ......
Take out n!/3!(n-3)!
\(\frac{n!}{3!(n-3)!}(1+\frac{n-3}{4} )=\frac{n!}{3!(n-3)!}(\frac{n+1}{4} ).....\frac{n+1}{4}=\frac{n+1}{4}\)....Always true..
This also tells us why nC3+nC4=(n+1)C4

A
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If n > 4, what is the value of the integer n ? [#permalink] New post   05 Mar 2022, 00:28
how is (N-3)! = (N-3).(N-4)! ?
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Re: If n > 4, what is the value of the integer n ? [#permalink] New post   08 Jun 2023, 08:59
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DV221 wrote:
how is (N-3)! = (N-3).(N-4)! ?


(N-3)! can be written as (N-3) (N-4) (N-5)... until you reach 1, whatever the value of N is.
Here, you will notice if you ignore (N-3) for a minute, the rest of the expansion denotes, in fact, the expansion for (N-4)!.
And hence, (N-3)! = (N-3) (N-4)!
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Re: If n > 4, what is the value of the integer n ? [#permalink]
08 Jun 2023, 08:59
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