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# Thirty-six identical chairs must be arranged in rows with the same num

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Math Expert
Joined: 02 Sep 2009
Posts: 56307
Thirty-six identical chairs must be arranged in rows with the same num  [#permalink]

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02 Nov 2017, 23:27
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Difficulty:

55% (hard)

Question Stats:

56% (01:44) correct 44% (01:36) wrong based on 132 sessions

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Thirty-six identical chairs must be arranged in rows with the same number of chairs in each row. Each row must contain at least 3 chairs, and there must be at least 3 rows. A row is parallel to the front of the room. How many different arrangements are possible?

(A) 2
(B) 4
(C) 5
(D) 6
(E) 10

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Re: Thirty-six identical chairs must be arranged in rows with the same num  [#permalink]

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02 Nov 2017, 23:34
1
Bunuel wrote:
Thirty-six identical chairs must be arranged in rows with the same number of chairs in each row. Each row must contain at least 3 chairs, and there must be at least 3 rows. A row is parallel to the front of the room. How many different arrangements are possible?

(A) 2
(B) 4
(C) 5
(D) 6
(E) 10

r>=3, c>=3
(r,c) = (3,12), (4,9), (6,6), (9,4), (12,3)

C
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Re: Thirty-six identical chairs must be arranged in rows with the same num  [#permalink]

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03 Nov 2017, 03:10
Rows Seats
3 12
4 9
6 6
9 4
12 3

Manager
Joined: 21 Jun 2014
Posts: 62
Re: Thirty-six identical chairs must be arranged in rows with the same num  [#permalink]

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03 Nov 2017, 04:23
1
36 chairs.. so what we are looking for is number of factors of 36 = 9. Out of this we need to remove cases in which one of the factor is less than 3 i.e (36,1) (18,2) so 9-4 = 5

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Joined: 29 Oct 2017
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Re: Thirty-six identical chairs must be arranged in rows with the same num  [#permalink]

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04 Nov 2017, 09:46
The solution can be analysed as m*n type of arrangements which are
12*3
3*12
6*6
4*9
9*4
given the total type of arrangement
Please correct me if i am wrong
Also Suggest any shortcut method to reduce time

Thanks
Math Expert
Joined: 02 Aug 2009
Posts: 7763
Thirty-six identical chairs must be arranged in rows with the same num  [#permalink]

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04 Nov 2017, 10:17
Bunuel wrote:
Thirty-six identical chairs must be arranged in rows with the same number of chairs in each row. Each row must contain at least 3 chairs, and there must be at least 3 rows. A row is parallel to the front of the room. How many different arrangements are possible?

(A) 2
(B) 4
(C) 5
(D) 6
(E) 10

hi..
Shubhashri10, you are looking for a shortcut, then here is a method if you analyze what the Q is all about..

the rows and columns are nothing but factors of 36.
$$36=2^2*3^2$$... so number of factors are 3*3=9

but the rows and columns are ATLEAST 3..
so take out factors which are less than 3 ;- 1 and 2, so TWO
but these 2 can take value of rows or columns so TWO ways

total 2*2=4

ans = 9-4=5
C
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Re: Thirty-six identical chairs must be arranged in rows with the same num  [#permalink]

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04 Nov 2017, 11:25
chetan2u wrote:
Bunuel wrote:
Thirty-six identical chairs must be arranged in rows with the same number of chairs in each row. Each row must contain at least 3 chairs, and there must be at least 3 rows. A row is parallel to the front of the room. How many different arrangements are possible?

(A) 2
(B) 4
(C) 5
(D) 6
(E) 10

hi..
Shubhashri10, you are looking for a shortcut, then here is a method if you analyze what the Q is all about..

the rows and columns are nothing but factors of 36.
$$36=2^2*3^2$$... so number of factors are 383=9

but the rows and columns are ATLEAST 3..
so take out factors which are less than 3 ;- 1 and 2, so TWO
but these 2 can take value of rows or columns so TWO ways

total 2*2=4

ans = 9-4=5
C

Hi,

Thank you
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Re: Thirty-six identical chairs must be arranged in rows with the same num  [#permalink]

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06 Nov 2017, 17:32
1
Bunuel wrote:
Thirty-six identical chairs must be arranged in rows with the same number of chairs in each row. Each row must contain at least 3 chairs, and there must be at least 3 rows. A row is parallel to the front of the room. How many different arrangements are possible?

(A) 2
(B) 4
(C) 5
(D) 6
(E) 10

We could have the following arrangements:

3 rows of 12 chairs per row

4 rows of 9 chairs per row

6 rows of 6 chairs per row

9 rows of 4 chairs per row

12 rows of 3 chairs per row

We see that the number of arrangements is 5.

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Posts: 7763
Re: Thirty-six identical chairs must be arranged in rows with the same num  [#permalink]

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06 Nov 2017, 19:33
TaN1213 wrote:
chetan2u wrote:
Bunuel wrote:
Thirty-six identical chairs must be arranged in rows with the same number of chairs in each row. Each row must contain at least 3 chairs, and there must be at least 3 rows. A row is parallel to the front of the room. How many different arrangements are possible?

(A) 2
(B) 4
(C) 5
(D) 6
(E) 10

hi..
Shubhashri10, you are looking for a shortcut, then here is a method if you analyze what the Q is all about..

the rows and columns are nothing but factors of 36.
$$36=2^2*3^2$$... so number of factors are 383=9

but the rows and columns are ATLEAST 3..
so take out factors which are less than 3 ;- 1 and 2, so TWO
but these 2 can take value of rows or columns so TWO ways

total 2*2=4

ans = 9-4=5
C

Hi,

Thank you

hi..
sorry for the error..
it is 3*3=9
basically (2+1)(2+1)
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Re: Thirty-six identical chairs must be arranged in rows with the same num  [#permalink]

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24 Jan 2019, 08:40
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Re: Thirty-six identical chairs must be arranged in rows with the same num   [#permalink] 24 Jan 2019, 08:40
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