Jul 16 03:00 PM PDT  04:00 PM PDT Join a free live webinar and find out which skills will get you to the top, and what you can do to develop them. Save your spot today! Tuesday, July 16th at 3 pm PST Jul 16 08:00 PM EDT  09:00 PM EDT Strategies and techniques for approaching featured GMAT topics. Tuesday, July 16th at 8 pm EDT Jul 15 10:00 PM PDT  11:00 PM PDT Get the complete Official GMAT Exam Pack collection worth $100 with the 3 Month Pack ($299) Jul 15 10:00 PM PDT  11:00 PM PDT Take 20% off the plan of your choice, now through midnight on Monday, July 15th Jul 19 08:00 AM PDT  09:00 AM PDT The Competition Continues  Game of Timers is a teambased competition based on solving GMAT questions to win epic prizes! Starting July 1st, compete to win prep materials while studying for GMAT! Registration is Open! Ends July 26th Jul 21 07:00 AM PDT  09:00 AM PDT Attend this webinar to learn a structured approach to solve 700+ Number Properties question in less than 2 minutes
Author 
Message 
TAGS:

Hide Tags

Current Student
Joined: 28 Dec 2004
Posts: 3092
Location: New York City
Schools: Wharton'11 HBS'12

In how many ways can 5 identical black balls and 7 identical
[#permalink]
Show Tags
Updated on: 21 Jun 2015, 11:07
Question Stats:
55% (01:56) correct 45% (02:49) wrong based on 244 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
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by FN on 22 Oct 2005, 09:49.
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.




Math Expert
Joined: 02 Sep 2009
Posts: 56233

Re: In how many ways can 5 identical black balls and 7 identical
[#permalink]
Show Tags
21 Jun 2015, 11:37
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 \(C^5_8=56\). Answer: A. Check other Seating Arrangements in a Row and around a Table questions in our Special Questions Directory.
_________________




Director
Joined: 21 Aug 2005
Posts: 731

Re: In how many ways can 5 identical black balls and 7 identical
[#permalink]
Show Tags
22 Oct 2005, 12:18
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)



VP
Joined: 30 Sep 2004
Posts: 1376
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.



Manager
Joined: 21 Aug 2012
Posts: 128
Concentration: General Management, Operations

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.



Manager
Joined: 03 Sep 2014
Posts: 75
Concentration: Marketing, Healthcare
Schools: Kellogg 1YR '17, Booth '16, McCombs '18, Tepper '18, INSEAD Jan '17, ISB '17, NUS '18, IIMA , IIMB, IIMC , IIML '15

In how many ways can 5 identical black balls and 7 identical
[#permalink]
Show Tags
21 Jun 2015, 11:56
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 Hence answer is A



SVP
Joined: 26 Mar 2013
Posts: 2283

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



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: 280
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 \(C^5_8=56\). Answer: A. 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!!



Math Expert
Joined: 02 Aug 2009
Posts: 7764

Re: In how many ways can 5 identical black balls and 7 identical
[#permalink]
Show Tags
27 Dec 2015, 04:35
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 \(C^5_8=56\). Answer: A. 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
_________________



Intern
Joined: 31 Oct 2015
Posts: 31

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_20160115152857.png.png [ 165.4 KiB  Viewed 3511 times ]



Intern
Joined: 25 May 2014
Posts: 35

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 \(C^5_8=56\). Answer: A. 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!



Manager
Joined: 23 Jan 2016
Posts: 180
Location: India
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.



Math Expert
Joined: 02 Sep 2009
Posts: 56233

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.
_________________



Manager
Joined: 20 Apr 2014
Posts: 88

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. please help.



Manager
Joined: 03 May 2015
Posts: 183
Location: South Africa
Concentration: International Business, Organizational Behavior
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. please help. Hi hatemnagLet 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 Cr1 = (3 + 61)C(61) = 8C5 = 56 If you have any doubt about the formula pm me, I will clarify the formula for you
_________________
Kudos if I helped



NonHuman User
Joined: 09 Sep 2013
Posts: 11649

Re: In how many ways can 5 identical black balls and 7 identical
[#permalink]
Show Tags
01 Apr 2019, 10:16
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.
_________________




Re: In how many ways can 5 identical black balls and 7 identical
[#permalink]
01 Apr 2019, 10:16






