Question: 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.)?

1. 10
2. 18
3. 26
4. 32
5. 64

Solution and Explanation:

Approach Solution (1):

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 hands played there could be the following outcomes:

WWWLL = \(\frac{5!}{3!*2!}\) = 10 ways this to occur;

WWLLL = \(\frac{5!}{3!*2!}\) = 10 ways this to occur;

WLLLL =\(\frac{5!}{4!*1!}\)= 5 ways this to occur

LLLLL = only 1 way this to occur

Total = 10 + 10 + 5 +1 = 26

Correct Option: C

Approach Solution (2):

If X is the number of wins and Y is the number 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 the first 5, we can have the following:
0 wins 5 losses = 1 way
1 win 4 losses = 5C1 = 5
2 wins 3 losses = 5C2 = 10
3 wins 2 losses = 5C3 = 10
And total ways = 26 ways

Correct Option: C

Approach Solution (3):

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.
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

Correct Option: C

“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.)?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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