As promised, below is the answer to our weekly Challenge Problem. Enjoy!
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.)
(A) 10
(B) 18
(C) 26
(D) 32
(E) 64
Answer
Let W be the number of wins and L be the number of losses. Since the total number of hands equals 12 and the net winnings equal $210, we can construct and solve the following simultaneous equations:
So we know that the gambler won 3 hands and lost 9. We do not know where in the sequence of 12 hands the 3 wins appear. So when counting the possible outcomes for the first 5 hands, we must consider these possible scenarios:
1) Three wins and two losses
2) Two wins and three losses
3) One win and four losses
4) No wins and five losses
In the first scenario, we have WWWLL. We need to know in how many different ways we can arrange these five letters:
So there are 10 possible arrangements of 3 wins and 2 losses.
The second scenario — WWLLL — will yield the same result: 10.
The third scenario — WLLLL — will yield 5 possible arrangements, since the one win has only 5 possible positions in the sequence.
The fourth scenario — LLLLL — will yield only 1 possible arrangement, since rearranging these letters always yields the same sequence.
Altogether, then, there are 10 + 10 + 5 + 1 = 26 possible outcomes for the gambler’s first five hands.
The correct answer is C.
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