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# In how many different ways can all of 5 identical balls be placed

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Joined: 27 May 2014
Posts: 525
GMAT 1: 730 Q49 V41
In how many different ways can all of 5 identical balls be placed  [#permalink]

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02 Feb 2017, 18:44
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00:00

Difficulty:

75% (hard)

Question Stats:

56% (02:28) correct 44% (02:42) wrong based on 95 sessions

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In how many different ways can all of 5 identical balls be placed in the cells shown in the figure such that each row contains at least 1 ball?
(A) 64
(B) 81
(C) 84
(D) 96
(E) 108

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File comment: Figure 1

Figure 1.JPG [ 10.48 KiB | Viewed 3189 times ]

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Re: In how many different ways can all of 5 identical balls be placed  [#permalink]

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03 Feb 2017, 02:56
2
3
saswata4s wrote:
In how many different ways can all of 5 identical balls be placed in the cells shown in the figure such that each row contains at least 1 ball?
(A) 64
(B) 81
(C) 84
(D) 96
(E) 108

Hi,

Another way..

1) Total ways of selecting 5 places out of 9 = 9C5=$$\frac{9!}{5!4!}=126$$..

2) ways in which the 5 places are in just two rows..
Selecting 2 rows out of 3=3C2=3..
These two rows have 6 cells, so choosing 5 out of 6 cells=6C5=6..
So total ways in which the balls are NOT in all three= 3*6=18..

Ways in which the balls are in all three cells=126-18=108

E
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##### General Discussion
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Re: In how many different ways can all of 5 identical balls be placed  [#permalink]

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02 Feb 2017, 20:15
2
The balls can be placed in the rows in 2 ways --> 3 - 1 - 1 or 2 - 2 - 1

Number of different ways = (3C3*3C1*3C1)*3 + (3C2*3C2*3C1)*3 = 27 + 81 = 108 ways

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Re: In how many different ways can all of 5 identical balls be placed  [#permalink]

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29 Dec 2017, 10:26
It is not mentioned in the ques that one cell can have only one ball. Both the solution mentioned above are based on this assumption. How did we assume it? Please help.. This assumptions are killing me
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Re: In how many different ways can all of 5 identical balls be placed  [#permalink]

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30 Dec 2017, 00:43
AR15J wrote:
It is not mentioned in the ques that one cell can have only one ball. Both the solution mentioned above are based on this assumption. How did we assume it? Please help.. This assumptions are killing me

hi AR15J

it is clear from the image that there are 3 rows. It is mentioned in the question that each row has at least 1 ball. So you have 5 balls to play with and 3 rows how are you going to distribute them?
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Re: In how many different ways can all of 5 identical balls be placed  [#permalink]

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30 Dec 2017, 02:48
niks18 wrote:
AR15J wrote:
It is not mentioned in the ques that one cell can have only one ball. Both the solution mentioned above are based on this assumption. How did we assume it? Please help.. This assumptions are killing me

hi AR15J

it is clear from the image that there are 3 rows. It is mentioned in the question that each row has at least 1 ball. So you have 5 balls to play with and 3 rows how are you going to distribute them?

Thanks Niks for your reply.. I understand this.. But why can't we place more than one ball in a cell. For example , to calculate -in how many ways 2 balls can be place in a row.. why the no of ways are -- 3c1*3c2? (selecting a row and then selecting two columns from that row to place two balls)

Why are we not considering the case when both the ball are in the same cell of a row? If we consider this, total cases will be 3c1*3c2+3c1
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Re: In how many different ways can all of 5 identical balls be placed  [#permalink]

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30 Dec 2017, 03:00
AR15J wrote:
niks18 wrote:
AR15J wrote:
It is not mentioned in the ques that one cell can have only one ball. Both the solution mentioned above are based on this assumption. How did we assume it? Please help.. This assumptions are killing me

hi AR15J

it is clear from the image that there are 3 rows. It is mentioned in the question that each row has at least 1 ball. So you have 5 balls to play with and 3 rows how are you going to distribute them?

Thanks Niks for your reply.. I understand this.. But why can't we place more than one ball in a cell. For example , to calculate -in how many ways 2 balls can be place in a row.. why the no of ways are -- 3c1*3c2? (selecting a row and then selecting two columns from that row to place two balls)

Why are we not considering the case when both the ball are in the same cell of a row? If we consider this, total cases will be 3c1*3c2+3c1

Hi AR15J

ohhhk now I got your point. So ideally a standard GMAT official question would have stated clearly that 1 cell contains 1 ball only. The solutions above have also assumed that 1 cell contains 1 ball only. As this is a question from an unknown source, i would suggest let's not think too much into that

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Re: In how many different ways can all of 5 identical balls be placed  [#permalink]

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07 Jan 2018, 06:15
AR15J wrote:
niks18 wrote:
AR15J wrote:
It is not mentioned in the ques that one cell can have only one ball. Both the solution mentioned above are based on this assumption. How did we assume it? Please help.. This assumptions are killing me

hi AR15J

it is clear from the image that there are 3 rows. It is mentioned in the question that each row has at least 1 ball. So you have 5 balls to play with and 3 rows how are you going to distribute them?

Thanks Niks for your reply.. I understand this.. But why can't we place more than one ball in a cell. For example , to calculate -in how many ways 2 balls can be place in a row.. why the no of ways are -- 3c1*3c2? (selecting a row and then selecting two columns from that row to place two balls)

Why are we not considering the case when both the ball are in the same cell of a row? If we consider this, total cases will be 3c1*3c2+3c1

what is more, it is stated that EACH ROW NOT EACH CELL must contain at least 1 ball

so we can first place 1 ball in each row, as there are 3 rows, we are left with (5 - 3) = 2 balls

now, as the rows and balls are identical, we can place the 2 balls in 3 rows as under

4!
____
2!2!

= 6

????
Re: In how many different ways can all of 5 identical balls be placed &nbs [#permalink] 07 Jan 2018, 06:15
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