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Math Expert V
Joined: 02 Sep 2009
Posts: 58452
There are 6 boxes numbered 1, 2, ... 6. Each box is to be filled up ei  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 65% (02:01) correct 35% (02:25) wrong based on 97 sessions

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There are 6 boxes numbered 1, 2, ... 6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is:

A. 5
B. 6
C. 21
D. 33
E. 60

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Intern  B
Joined: 16 Apr 2015
Posts: 31
There are 6 boxes numbered 1, 2, ... 6. Each box is to be filled up ei  [#permalink]

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Bunuel wrote:
There are 6 boxes numbered 1, 2, ... 6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is:

A. 5
B. 6
C. 21
D. 33
E. 60

6 Green - 1 way
5 green - 2 ways
4 green - 3 ways
3 green - 4 ways
2 green - 5 ways
1 green - 6 ways

total = > 21 ways
Director  S
Joined: 17 Dec 2012
Posts: 626
Location: India
Re: There are 6 boxes numbered 1, 2, ... 6. Each box is to be filled up ei  [#permalink]

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For 1 G one case is:
GRRRRR. There are 6 possibilities in this with G occupying box 1 or 2 or... 6.

For 2 Gs, we can think of another case:
GGRRRR: This has 5 possibilities

And for 3Gs, 4Gs, 5Gs and 6Gs we have 4,3,2, and 1 possibilities
A total of 6+5+4+3+2+1=21
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There are 6 boxes numbered 1, 2 ,3, 4, 5, 6 Each box is to be filled u  [#permalink]

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There are 6 boxes numbered 1, 2 ,3, 4, 5, 6 Each box is to be filled up either with red or green ball in a such way that at lest 1 box contains green ball and the boxes containing green balls are consecutively numbered. the total number of ways in which this can be done is

A 5
B 21
C 33
D 60
E 40
Current Student B
Status: Persevere
Joined: 09 Jan 2016
Posts: 117
Location: Hong Kong
GMAT 1: 750 Q50 V41 GPA: 3.52
Re: There are 6 boxes numbered 1, 2 ,3, 4, 5, 6 Each box is to be filled u  [#permalink]

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satishchaudharygmat wrote:
There are 6 boxes numbered 1, 2 ,3, 4, 5, 6 Each box is to be filled up either with red or green ball in a such way that at lest 1 box contains green ball and the boxes containing green balls are consecutively numbered. the total number of ways in which this can be done is

A 5
B 21
C 33
D 60
E 40

Hi!
The topic should contain the first few words of the question. Requesting the author and the moderators to fix the issue.

As far as the question is concerned:
If there is one box that contains a green ball then there will be 6 ways to do it: 1, 2, 3, 4, 5, or 6
If there are two boxes that contain green ball(s) then there will be 5 ways to do it: (1,2) (2,3) (3,4) (4,5) or (5,6)
If there are three boxes that contain green ball(s) then there will be 4 ways to do it: (1,2,3) (2,3,4) (3,4,5) or (4,5,6)
If there are four boxes that contain green ball(s) then there will be 3 ways to do it: (1,2,3,4) (2,3,4,5) or (3,4,5,6)
If there are five boxes that contain green ball(s) then there will be 2 ways to do it: (1,2,3,4,5) or (2,3,4,5,6)
If there are six boxes that contain green ball(s) then there will be 1 way to do it: (1,2,3,4,5,6)

So in total 1+2+3+4+5+6 = 21 ways
Current Student Joined: 18 Oct 2014
Posts: 803
Location: United States
GMAT 1: 660 Q49 V31 GPA: 3.98
Re: There are 6 boxes numbered 1, 2 ,3, 4, 5, 6 Each box is to be filled u  [#permalink]

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1
satishchaudharygmat wrote:
There are 6 boxes numbered 1, 2 ,3, 4, 5, 6 Each box is to be filled up either with red or green ball in a such way that at lest 1 box contains green ball and the boxes containing green balls are consecutively numbered. the total number of ways in which this can be done is

A 5
B 21
C 33
D 60
E 40

Ways to put green ball in only one box= 6
Ways to put green ball in 2 consecutively numbered boxes= 5
Ways to put green ball in 3 consecutively numbered boxes= 4
Ways to put green ball in 4 consecutively numbered boxes= 3
Ways to put green ball in 5 consecutively numbered boxes= 2
Ways to put green ball in 6 consecutively numbered boxes= 1

Total ways= 21

B is the answer
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Re: There are 6 boxes numbered 1, 2, ... 6. Each box is to be filled up ei  [#permalink]

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1
Various ways are as under.

GGGGGG - 1 way

GGGGGR: Bundle all 5 G's together. This bundle and R have to be placed. Can only be done in 2 ways

GGGGRR: Bundle all 4 G's together. This bundle and 2 R's have to be distributed. Total 3 articles are there. This can be done in (3!/2!= 3 ways)

[There is a bundle of G's and 2 R's are similar. That is why we are dividing by 2!]

GGGRRR: Bundle all 3 G's together. This bundle and 3 R's have to be distributed. Total 4 articles are there. This can be done in (4!/3!= 4 ways)

[There is a bundle of G's and 3 R's are similar. That is why we are dividing by 3!]

GGRRRR: Bundle 2 G's together. This bundle and 4 R's have to be distributed. Total 5 articles are there. This can be done in (5!/4!= 5 ways)

[There is a bundle of G's and 4 R's are similar. That is why we are dividing by 4!]

GRRRRR: Now there is only one G. This can be kept anywhere. 6 articles are there, 5 of one type 1 of a different type. These can be arranged in (6!/5!= 6 ways)

so 1+2+3+4+5+6 = 21 ways

Option B is correct.
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Joined: 15 Jan 2018
Posts: 6
Re: There are 6 boxes numbered 1, 2, ... 6. Each box is to be filled up ei  [#permalink]

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Bunuel wrote:
There are 6 boxes numbered 1, 2, ... 6. Each box is to be filled up either with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is:

A. 5
B. 6
C. 21
D. 33
E. 60

Hi Bunuel,

Why can't we apply the repetition method/formula as below:
GRRRRR is 6!/5!1! = 6 ways
GGRRRR is 6!/2!4! = 15 ways
GGGRRR is 6!/3!3! = 20 ways and so on ...

I understand that the method gives correct answer if used as below:
GRRRRR is 6!/5!1! = 6 ways
GGRRRR is 5!/4! = 5 ways and so on ...

G color since there are placed consecutive have to be considered as one unit.
Just wanted to clarify if my understanding is right.

Thanks in advance for the response!! Re: There are 6 boxes numbered 1, 2, ... 6. Each box is to be filled up ei   [#permalink] 16 Mar 2018, 05:07
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