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Difference between revisions of "2014 AIME II Problems/Problem 9"

(Created page with "==Solution== Starting with small cases, we see that four chairs give 4 + 1 = 5, five chairs give 5 + 5 + 1 = 11, and six chairs give 6 + 6 + 6 + 6 + 1 = 25. Thus, I claim that n ...")
 
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==Solution==
 
==Solution==
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Starting with small cases, we see that four chairs give 4 + 1 = 5, five chairs give 5 + 5 + 1 = 11, and six chairs give 6 + 6 + 6 + 6 + 1 = 25. Thus, I claim that n chairs give <math>n 2^{n-4} + 1</math>, as confirmed above. This claim can be verified by PIE: there are <math>n 2^(n-3}</math> ways to arrange 3 adjacent chairs, but then we subtract <math>n 2^{n-4}</math> ways to arrange 4. Finally, we add 1 to account for all chairs case (the other cases have been taken care of). Thus, for n = 10 we get <math>\boxed{641}</math>.
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Starting with small cases, we see that four chairs give 4 + 1 = 5, five chairs give 5 + 5 + 1 = 11, and six chairs give 6 + 6 + 6 + 6 + 1 = 25. Thus, I claim that n chairs give <math>n 2^{n-4} + 1</math>, as confirmed above. This claim can be verified by PIE: there are <math>n 2^{n-3}</math> ways to arrange 3 adjacent chairs, but then we subtract <math>n 2^{n-4}</math> ways to arrange 4. Finally, we add 1 to account for all chairs case (the other cases have been taken care of). Thus, for n = 10 we get <math>\boxed{641}</math>.

Revision as of 18:18, 28 March 2014

Solution

Starting with small cases, we see that four chairs give 4 + 1 = 5, five chairs give 5 + 5 + 1 = 11, and six chairs give 6 + 6 + 6 + 6 + 1 = 25. Thus, I claim that n chairs give $n 2^{n-4} + 1$, as confirmed above. This claim can be verified by PIE: there are $n 2^{n-3}$ ways to arrange 3 adjacent chairs, but then we subtract $n 2^{n-4}$ ways to arrange 4. Finally, we add 1 to account for all chairs case (the other cases have been taken care of). Thus, for n = 10 we get $\boxed{641}$.

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