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# A gambler began playing blackjack with $110 in chips. After  Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: GMAT Club Legend Joined: 07 Jul 2004 Posts: 5063 Location: Singapore Followers: 26 Kudos [?]: 237 [5] , given: 0 A gambler began playing blackjack with$110 in chips. After [#permalink]  22 Mar 2005, 03:01
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A gambler began playing blackjack with $110 in chips. After exactly 12 hands, he left the table with$320 in chips, having won some hands and lost others. Each win earned $100 and each loss cost$10. How many possible outcomes were there for the first 5 hands he played? (For example, won the first hand, lost the second, etc.)

(A) 10
(B) 18
(C) 26
(D) 32
(E) 64
[Reveal] Spoiler: OA
Senior Manager
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I guess! (C)
well the algebra first -> 110 (starting cash)+100x (n of wins)-10*(12-x)=320 -> solve and get x=3
if wins=3/12
if losses= 9/12
there can be max 3 wins out of 5 hands
alas, I'm not sure about how to turn out the outcomes
I think
5c0+5c1+5c2+5c3=26
that is 0 wins/1 win/2 wins/3 wins
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I got the algebra bit correct too: 3 Wins and 9 losses.

I'm unsure on how to go about the arranging part. thearch, I guess your approach looks fine and C should be the answer.

Good question, ywilfred!
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i would have to go with C also because he can only have 3 wins and 9losses in 12 rounds
so in 5 turns the possibilities are limited to getting no wins, 1 win, 2 wins and all 3 wins.
26
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"C"

If X is number of wins and Y num of losses then

100X - 10Y = 210

10X-Y = 21......only when Y = 9 and X = 3 it satisfies....so we have 3 wins and 9 losses.

for first 5, we can have the following:

0 wins 5 losses = 1 way
1 win and 4 losses = 5C1 = 5
2 wins and 3 losses = 5C2 = 10
3 wins and 2 losses = 5C3 = 10

Add total ways = 26 ways
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Good job thearch. You can also do it this way: 2^5-5(four wins)-1(five wins)=26.
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Re: A gambler began playing blackjack with $110 in chips. After [#permalink] 24 Feb 2014, 02:18 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. _________________ Intern Joined: 17 Sep 2013 Posts: 7 Followers: 0 Kudos [?]: 0 [0], given: 18 Re: [#permalink] 11 Mar 2014, 09:28 What about arrangement of these outcomes, don't we have to arrange their sequence? I applied permutation instead of combination, got the wrong answer. Kindly help Thanks! Math Expert Joined: 02 Sep 2009 Posts: 31356 Followers: 5367 Kudos [?]: 62595 [5] , given: 9457 Re: Re: [#permalink] 11 Mar 2014, 09:59 5 This post received KUDOS Expert's post 2 This post was BOOKMARKED ShantnuMathuria wrote: What about arrangement of these outcomes, don't we have to arrange their sequence? I applied permutation instead of combination, got the wrong answer. Kindly help Thanks! A gambler began playing blackjack with$110 in chips. After exactly 12 hands, he left the table with $320 in chips, having won some hands and lost others. Each win earned$100 and each loss cost $10. How many possible outcomes were there for the first 5 hands he played? (For example, won the first hand, lost the second, etc.) (A) 10 (B) 18 (C) 26 (D) 32 (E) 64 The gambler started with$110 and left with $320, thus he/she in 12 hands won$320 - $110 =$210:

100W - 10L = 210;
100W - 10(12-W) = 210 (since Wins + Loss = 12) --> W = 3.

So, we have that out of 12 hands the gambler won 3 hands and lost 9.

For the first 5 hands played there could be the following outcomes:
WWWLL --> 5!/(3!2!) = 10 ways this to occur (for example, WWWLL, WWLWL, WLWWL, ...);
WWLLL --> 5!/(3!2!) = 10 ways this to occur;
WLLLL --> 5!/(4!1!) = 5 ways this to occur;
LLLLL --> only 1 way this to occur.

Total = 10 + 10 + 5 + 1 = 26.

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Re: A gambler began playing blackjack with $110 in chips. After [#permalink] 01 Apr 2014, 19:44 Can someone explain where I can learn more about this: WWWLL --> 5!/(3!2!) = 10 Why do you divide here. I think i get the logic, but how do you know to choose 3! and 2!, and how can I know when to do this. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 6238 Location: Pune, India Followers: 1683 Kudos [?]: 9672 [1] , given: 198 Re: A gambler began playing blackjack with$110 in chips. After [#permalink]  01 Apr 2014, 20:11
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Expert's post
frenchwr wrote:
Can someone explain where I can learn more about this: WWWLL --> 5!/(3!2!) = 10 Why do you divide here. I think i get the logic, but how do you know to choose 3! and 2!, and how can I know when to do this.

You arrange 5 distinct objects in 5! ways.

But if some of them are identical, you need to divide the total arrangements by the factorial of that number: Say you have total n objects out of which m are identical.
Total number of arrangements = n!/m!

e.g. Out of 5 objects, if 2 are identical, number of arrangements = 5!/2! (because we don't have as many arrangements as before now.)

Say 5 objects are A, B, C, D and D. There are 2 identical Ds.
5! gives the arrangements of 5 distinct objects(e.g. ABCDE, ABCED are two diff arrangements) but if two letters are same, ABCDD is same as ABCDD (we flipped the D with the other D). Hence the number of arrangements are half in this case: 5!/2!

Similarly, if you have 5 letters such that three of them are same and another 2 are same, the number of arrangements is given by 5!/(3!*2!) as is the case with WWWLL.

Our Combinatorics book discusses this concept as well as other GMAT relevant concepts in detail. You can take a look at it here: http://www.amazon.com/Veritas-Prep-Stat ... ds=veritas
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 10 May 2015 Posts: 30 Followers: 0 Kudos [?]: 1 [0], given: 268 Re: A gambler began playing blackjack with$110 in chips. After [#permalink]  24 May 2015, 09:13
Guys...............I have a strange doubt about this..okay we so know there are 3 wins and 9 losses..and to have 5 hands..you could select 3 wins and 2 losses say..now why can't we just do 9c2(2losses out of 9)* 3c3(3wins out of 3).If we say it's identical then you can divide it by 3 ! and 2! respectively.However,that's still miserably wrong..I know it's wrong but can't figure out why? Please help!!!
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Re: A gambler began playing blackjack with $110 in chips. After [#permalink] 24 May 2015, 18:55 2 This post received KUDOS Expert's post davesinger786 wrote: Guys...............I have a strange doubt about this..okay we so know there are 3 wins and 9 losses..and to have 5 hands..you could select 3 wins and 2 losses say..now why can't we just do 9c2(2losses out of 9)* 3c3(3wins out of 3).If we say it's identical then you can divide it by 3 ! and 2! respectively.However,that's still miserably wrong..I know it's wrong but can't figure out why? Please help!!! You cannot select losses out of losses - they are all just losses. You can select hands to which you will allot losses since the hands are distinct - first hand, second hand .. till 12th hand. For example, if we have to give 3 wins and 2 losses to 5 hands, we can select the 2 hands to which we will give losses. We can do this in 5C2 ways = 10 ways. The other 3 hands will automatically be left with wins. This is another way of doing what Bunuel did above. Similarly, to give 3 losses we select 3 hands out of 5 in 5C3 ways = 10 ways To give 4 losses, we select 4 hands out of 5 in 5C4 = 5 ways To give 5 losses, we select 5 hands out of 5 in 5C5 = 1 way Total = 10+10+5+1 = 26 _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: A gambler began playing blackjack with $110 in chips. After [#permalink] 24 May 2015, 19:14 Thank you Karishma for your reply. So ,in this case do you mean to say that since they're all identical,we can't select losses out of losses? Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 6238 Location: Pune, India Followers: 1683 Kudos [?]: 9672 [0], given: 198 Re: A gambler began playing blackjack with$110 in chips. After [#permalink]  24 May 2015, 19:43
Expert's post
davesinger786 wrote:
Thank you Karishma for your reply. So ,in this case do you mean to say that since they're all identical,we can't select losses out of losses?

Yes, think of it this way:
If you have 12 different houses and you have to paint 5 of them - 3 red and 2 yellow, can you select red out of red? You must select the 3 houses out of 5 which you will paint red.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 10 May 2015 Posts: 30 Followers: 0 Kudos [?]: 1 [0], given: 268 Re: A gambler began playing blackjack with$110 in chips. After [#permalink]  25 May 2015, 19:13
ah I see your point.Thank you for that,Karishma!
Re: A gambler began playing blackjack with $110 in chips. After [#permalink] 25 May 2015, 19:13 Similar topics Replies Last post Similar Topics: 9 Car X began traveling at an average speed of 35 miles per hour. After 7 02 Sep 2015, 21:50 3 In blackjack, each card is worth a certain amount of points. 3 30 Apr 2013, 20:37 2 The game of blackjack is played with a deck consisting of 6 27 Feb 2013, 03:54 14 Kay began a certain game with x chips. On each of the next 9 07 Jan 2013, 04:27 12 A gambler bought$3,000 worth of chips at a casino in 8 25 Feb 2012, 21:23
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