It is currently 19 Oct 2017, 09:37

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# In how many ways can 5 identical black balls and 7 identical

Author Message
TAGS:

### Hide Tags

Current Student
Joined: 28 Dec 2004
Posts: 3351

Kudos [?]: 319 [2], given: 2

Location: New York City
Schools: Wharton'11 HBS'12
In how many ways can 5 identical black balls and 7 identical [#permalink]

### Show Tags

22 Oct 2005, 09:49
2
KUDOS
21
This post was
BOOKMARKED
00:00

Difficulty:

75% (hard)

Question Stats:

57% (01:34) correct 43% (02:44) wrong based on 316 sessions

### HideShow timer Statistics

In how many ways can 5 identical black balls and 7 identical white balls be arranged in a row so that no 2 black balls are together

A. 56
B. 64
C. 65
D. 316
E. 560
[Reveal] Spoiler: OA

Last edited by Bunuel on 21 Jun 2015, 11:07, edited 1 time in total.
Renamed the topic, edited the question and added the OA.

Kudos [?]: 319 [2], given: 2

Director
Joined: 21 Aug 2005
Posts: 786

Kudos [?]: 28 [2], given: 0

Re: In how many ways can 5 identical black balls and 7 identical [#permalink]

### Show Tags

22 Oct 2005, 12:18
2
KUDOS
2
This post was
BOOKMARKED
fresinha12 wrote:
8C3? or 8C5?

gsr wrote:
A) 8C3 = 56

8C3 = 8C5
Position of white balls is fixed. Since they are identical, number of ways to arrange them is 1.
We have 6 spaces in btw the white balls and 1 each to the left and right end to accomodate the black ball. So 8 places in total to arrange 5 balls
8C5 (or 8C3)

Kudos [?]: 28 [2], given: 0

VP
Joined: 30 Sep 2004
Posts: 1480

Kudos [?]: 411 [0], given: 0

Location: Germany
Re: In how many ways can 5 identical black balls and 7 identical [#permalink]

### Show Tags

23 Oct 2005, 08:24
there are 8 spaces availale so that no two black balls are together. that means 8! arrangements. but there are 5 identical black balls and 3 identical spaces. that`s why we have to divide by 5! and 3!. 8!/(5!3!).
_________________

If your mind can conceive it and your heart can believe it, have faith that you can achieve it.

Kudos [?]: 411 [0], given: 0

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 16673

Kudos [?]: 273 [0], given: 0

Re: In how many ways can 5 identical black balls and 7 identical [#permalink]

### Show Tags

21 Jun 2015, 07:39
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Kudos [?]: 273 [0], given: 0

Manager
Joined: 21 Aug 2012
Posts: 188

Kudos [?]: 52 [0], given: 349

Concentration: General Management, Operations
Schools: HBS '19 (S)
GMAT 1: 740 Q49 V42
Re: In how many ways can 5 identical black balls and 7 identical [#permalink]

### Show Tags

21 Jun 2015, 09:18
Can anyone help me with this? I have no idea how to solve this.

Kudos [?]: 52 [0], given: 349

Math Expert
Joined: 02 Sep 2009
Posts: 41893

Kudos [?]: 128870 [2], given: 12183

Re: In how many ways can 5 identical black balls and 7 identical [#permalink]

### Show Tags

21 Jun 2015, 11:37
2
KUDOS
Expert's post
5
This post was
BOOKMARKED
roopika2990 wrote:
Can anyone help me with this? I have no idea how to solve this.

In how many ways can 5 identical black balls and 7 identical white balls be arranged in a row so that no 2 black balls are together?

A. 56
B. 64
C. 65
D. 316
E. 560

Consider the following arrangement:

*W*W*W*W*W*W*W*

Now, if we replace any 5 out of 8 stars with black balls, then no 2 black balls will be together.

# of ways to choose 5 out of 8 is $$C^5_8=56$$.

Check other Seating Arrangements in a Row and around a Table questions in our Special Questions Directory.
_________________

Kudos [?]: 128870 [2], given: 12183

Manager
Joined: 03 Sep 2014
Posts: 75

Kudos [?]: 48 [0], given: 89

Concentration: Marketing, Healthcare
In how many ways can 5 identical black balls and 7 identical [#permalink]

### Show Tags

21 Jun 2015, 11:56
1
This post was
BOOKMARKED
roopika2990 wrote:
Can anyone help me with this? I have no idea how to solve this.

Consider the possible arrangements:

WBWBWBWBWBWW or BWBWBWBWBWWW or WWBWBWBWBWBW or WWWBWBWBWBWB

Clearly, we are concerned about not placing 2 B's together, whites can be placed however we want.

So, it is like _W_W_W_W_W_W_W_ where black ball can be placed anywhere on the blank spaces, and this way, there never will be a case where 2 black balls fall together.

As there are 8 black spaces where 5 black balls can be placed, thus, the combination turns out as 8C5 (again, we are not concerned about the placement of white balls as we have already "fixed" their positions in the diagram above)

Also, we achieved at 8C5 as arranging 5 black balls on 8 different spots => 8!/5!3! (Total combination of available spaces = 8!; choosing 5 spaces out of them = 5!, arranging 3 leftover spaces = 3!) = 56

Kudos [?]: 48 [0], given: 89

VP
Joined: 26 Mar 2013
Posts: 1260

Kudos [?]: 285 [0], given: 163

Re: In how many ways can 5 identical black balls and 7 identical [#permalink]

### Show Tags

27 Jul 2015, 10:07
King407 wrote:
roopika2990 wrote:
Can anyone help me with this? I have no idea how to solve this.

Consider the possible arrangements:

WBWBWBWBWBWW or BWBWBWBWBWWW or WWBWBWBWBWBW or WWWBWBWBWBWB

Clearly, we are concerned about not placing 2 B's together, whites can be placed however we want.

So, it is like _W_W_W_W_W_W_W_ where black ball can be placed anywhere on the blank spaces, and this way, there never will be a case where 2 black balls fall together.

As there are 8 black spaces where 5 black balls can be placed, thus, the combination turns out as 8C5 (again, we are not concerned about the placement of white balls as we have already "fixed" their positions in the diagram above)

Also, we achieved at 8C5 as arranging 5 black balls on 8 different spots => 8!/5!3! (Total combination of available spaces = 8!; choosing 5 spaces out of them = 5!, arranging 3 leftover spaces = 3!) = 56

Hi King407,
Can you elaborate how there is 8 places although there are 5 black balls? I do not get it

Thanks

Kudos [?]: 285 [0], given: 163

Senior Manager
Status: Always try to face your worst fear because nothing GOOD comes easy. You must be UNCOMFORTABLE to get to your COMFORT ZONE
Joined: 15 Aug 2014
Posts: 358

Kudos [?]: 86 [0], given: 472

Concentration: Marketing, Technology
GMAT 1: 570 Q44 V25
GMAT 2: 600 Q48 V25
WE: Information Technology (Consulting)
Re: In how many ways can 5 identical black balls and 7 identical [#permalink]

### Show Tags

27 Dec 2015, 02:36
Bunuel wrote:
roopika2990 wrote:
Can anyone help me with this? I have no idea how to solve this.

In how many ways can 5 identical black balls and 7 identical white balls be arranged in a row so that no 2 black balls are together?

A. 56
B. 64
C. 65
D. 316
E. 560

Consider the following arrangement:

*W*W*W*W*W*W*W*

Now, if we replace any 5 out of 8 stars with black balls, then no 2 black balls will be together.

# of ways to choose 5 out of 8 is $$C^5_8=56$$.

Check other Seating Arrangements in a Row and around a Table questions in our Special Questions Directory.

Hi Bunuel,

don't we consider here arranging 7 white balls in 7 places as 7!

so that num of ways can 5 identical black balls and 7 identical white balls be arranged in a row so that no 2 black balls are together is 7!*56
_________________

"When you want to succeed as bad as you want to breathe, then you’ll be successful.” - Eric Thomas

I need to work on timing badly!!

Kudos [?]: 86 [0], given: 472

Math Forum Moderator
Joined: 02 Aug 2009
Posts: 4971

Kudos [?]: 5472 [1], given: 112

Re: In how many ways can 5 identical black balls and 7 identical [#permalink]

### Show Tags

27 Dec 2015, 04:35
1
KUDOS
Expert's post
smartguy595 wrote:
Bunuel wrote:
roopika2990 wrote:
Can anyone help me with this? I have no idea how to solve this.

In how many ways can 5 identical black balls and 7 identical white balls be arranged in a row so that no 2 black balls are together?

A. 56
B. 64
C. 65
D. 316
E. 560

Consider the following arrangement:

*W*W*W*W*W*W*W*

Now, if we replace any 5 out of 8 stars with black balls, then no 2 black balls will be together.

# of ways to choose 5 out of 8 is $$C^5_8=56$$.

Check other Seating Arrangements in a Row and around a Table questions in our Special Questions Directory.

Hi Bunuel,

don't we consider here arranging 7 white balls in 7 places as 7!

so that num of ways can 5 identical black balls and 7 identical white balls be arranged in a row so that no 2 black balls are together is 7!*56

Hi,
since the balls are identical, all 7 balls can be arranged at 7 places only in one way...
yes had the ball been not identical, there would have been 7! ways
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

Kudos [?]: 5472 [1], given: 112

Intern
Joined: 31 Oct 2015
Posts: 37

Kudos [?]: 6 [0], given: 53

Re: In how many ways can 5 identical black balls and 7 identical [#permalink]

### Show Tags

15 Jan 2016, 13:33
Here's a different way to solve it. Posted an incorrect response earlier but eventually figured it out.
Attachments

SmartSelectImage_2016-01-15-15-28-57.png.png [ 165.4 KiB | Viewed 2185 times ]

Kudos [?]: 6 [0], given: 53

Intern
Joined: 25 May 2014
Posts: 49

Kudos [?]: 14 [0], given: 59

Re: In how many ways can 5 identical black balls and 7 identical [#permalink]

### Show Tags

26 Mar 2016, 09:45
Bunuel wrote:
roopika2990 wrote:
Can anyone help me with this? I have no idea how to solve this.

In how many ways can 5 identical black balls and 7 identical white balls be arranged in a row so that no 2 black balls are together?

A. 56
B. 64
C. 65
D. 316
E. 560

Consider the following arrangement:

*W*W*W*W*W*W*W*

Now, if we replace any 5 out of 8 stars with black balls, then no 2 black balls will be together.

# of ways to choose 5 out of 8 is $$C^5_8=56$$.

Check other Seating Arrangements in a Row and around a Table questions in our Special Questions Directory.

Hi Bunnel

Can you please help me understand why can't we solve it using Total arrangements - Number of arrangements where blacck balls are always together

Total = 12!/(7!*5!)
Black balls together: 8!/7!

Kudos [?]: 14 [0], given: 59

Manager
Joined: 23 Jan 2016
Posts: 201

Kudos [?]: 21 [0], given: 508

Location: India
GMAT 1: 600 Q43 V31
GMAT 2: 710 Q48 V40
GPA: 3.2
Re: In how many ways can 5 identical black balls and 7 identical [#permalink]

### Show Tags

12 Apr 2016, 05:53
Hi,

Is it correct to say in this question that the arrangement of balls within their group does not matter in this question as the white balls and black balls are identical within their respective groups?

Also is there a way to answer is by the 'reduction from all possible arrangements' method? i.e 12! - something? I thought that could be possible as there is condition for the black balls to not be togher; hence we can maybe 'glue' them and reduce them from all possible arrangements. Not sure how to do that though.

Kudos [?]: 21 [0], given: 508

Math Expert
Joined: 02 Sep 2009
Posts: 41893

Kudos [?]: 128870 [0], given: 12183

Re: In how many ways can 5 identical black balls and 7 identical [#permalink]

### Show Tags

12 Apr 2016, 06:55
abypatra wrote:
Hi,

Is it correct to say in this question that the arrangement of balls within their group does not matter in this question as the white balls and black balls are identical within their respective groups?

Also is there a way to answer is by the 'reduction from all possible arrangements' method? i.e 12! - something? I thought that could be possible as there is condition for the black balls to not be togher; hence we can maybe 'glue' them and reduce them from all possible arrangements. Not sure how to do that though.

We are told that 5 black balls and 7 white are identical, so we are not bothered about their arrangements. As for your second question, yes, it's possible to do it with (total) - (restriction) approach but it will be much more tedious than the approach used above.
_________________

Kudos [?]: 128870 [0], given: 12183

Manager
Joined: 20 Apr 2014
Posts: 116

Kudos [?]: [0], given: 23

Re: In how many ways can 5 identical black balls and 7 identical [#permalink]

### Show Tags

20 Apr 2016, 05:19
Hi Bunuel

why we should start arrangement with black not white.
i guess if we start with white, it would be 7 black places.

Kudos [?]: [0], given: 23

Current Student
Joined: 03 May 2015
Posts: 262

Kudos [?]: 84 [0], given: 23

Location: South Africa
GPA: 3.49
WE: Web Development (Insurance)
Re: In how many ways can 5 identical black balls and 7 identical [#permalink]

### Show Tags

19 Jul 2016, 05:21
hatemnag wrote:
Hi Bunuel

why we should start arrangement with black not white.
i guess if we start with white, it would be 7 black places.

Hi hatemnag

Let me show you the way with black balls fixed

B B B B B

Now restriction is that no two black ball can be together

So 4 white balls fixed

B W B W B W B W B

Now 3 white balls remain. But remember, all are identical so it can't be 18C3

So spaces remaining

_ (BW) _ (BW) _ (BW) _ (BW) _ B _ = 6 spaces

Where each blank space can have 0 to 3 balls each

Now we have to divide 3 identical balls among 6 spaces, where each can be from 0 to 3

I hope you know the formula for dividing identical objects among groups :

n+ r -1 Cr-1 = (3 + 6-1)C(6-1) = 8C5 = 56

If you have any doubt about the formula pm me, I will clarify the formula for you
_________________

Kudos if I helped

Kudos [?]: 84 [0], given: 23

Re: In how many ways can 5 identical black balls and 7 identical   [#permalink] 19 Jul 2016, 05:21
Display posts from previous: Sort by