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felix2018

铁杆木虫 (正式写手)

[求助] 组合数学的大神进来看看!

n个元素排成一排,用红黄蓝三种颜色着色,其中红色出现偶数次的着色方法有多少种?

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hank612

至尊木虫 (著名写手)

【答案】应助回帖

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felix2018: 金币+2, ★★★很有帮助 2013-11-12 17:27:57
my answer is slightly different from Jabile's answer. Lou2 Zhu3 make your own decision.

Let Pn be the prob of occurrence of red color being even, then by consider the first element is red or not, we obtain that

P(n)= 1/3 * (1-P(n-1)) + 2/3* P(n-1) = 1/3 ( 1+ P(n-1)).
Clearly P(1) = 2/3. Therefore
P(n)= 1/2 * ( 1+ 3^{-n} ) can be checked directly.
We_must_know. We_will_know.
3楼2013-11-12 10:33:00
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jabile

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【答案】应助回帖

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felix2018: 金币+1, ★★★很有帮助 2013-11-12 17:27:23
红色出现偶数次的方法为
A=\sum_{k=1}^{[n/2]}  C_n^{2k}  2^{n-2k}
令B=\sum_{k=1}^{[n/2]} C_n^{2k+1} 2^{n-2k-1}
A+B=3^n, A-B=(1-2)^n
A=1/2  [3^n+(-1)^n]
2楼2013-11-10 12:23:40
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felix2018

铁杆木虫 (正式写手)

引用回帖:
3楼: Originally posted by hank612 at 2013-11-12 10:33:00
my answer is slightly different from Jabile's answer. Lou2 Zhu3 make your own decision.

Let Pn be the prob of occurrence of red color being even, then by consider the first element is red or not,  ...

谢谢3楼好心的你,简洁的做法!

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4楼2013-11-12 17:24:51
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