Difference between revisions of "2008 AIME I Problems/Problem 11"

Revision as of 21:42, 22 November 2012

Problem

Consider sequences that consist entirely of $A$ 's and $B$ 's and that have the property that every run of consecutive $A$ 's has even length, and every run of consecutive $B$ 's has odd length. Examples of such sequences are $AA$ , $B$ , and $AABAA$ , while $BBAB$ is not such a sequence. How many such sequences have length 14?

Solution

Solution 1

Let $a_n$ and $b_n$ denote, respectively, the number of sequences of length $n$ ending in $A$ and $B$ . If a sequence ends in an $A$ , then it must have been formed by appending two $A$ s to the end of a string of length $n-2$ . If a sequence ends in a $B,$ it must have either been formed by appending one $B$ to a string of length $n-1$ ending in an $A$ , or by appending two $B$ s to a string of length $n-2$ ending in a $B$ . Thus, we have the recursions $\begin{align*} a_n &= a_{n-2} + b_{n-2}\\ b_n &= a_{n-1} + b_{n-2} \end{align*}$ By counting, we find that $a_1 = 0, b_1 = 1, a_2 = 1, b_2 = 0$ .

\[\begin{tabular}{|r||r|r|||r||r|r|}
\hline
n & a_n & b_n & n & a_n & b_n\\
\hline
1&0&1&
8&6&10\\
2&1&0&
9&11&11\\
3&1&2&
10&16&21\\
4&1&1&
11&22&27\\
5&3&3&
12&37&43\\
6&2&4&
13&49&64\\
7&6&5&
14&80&92\\
\hline
\end{tabular}\] (Error compiling LaTeX. Unknown error_msg)

Therefore, the number of such strings of length $14$ is $a_{14} + b_{14} = \boxed{172}$ .

Solution 2

We replace "14" with " $2n$ ". We first note that we must have an even number of chunks of $B$ 's, because of parity issues. We then note that every chunk of $B$ 's except the last one must end in the sequence $BAA$ , since we need at least two $A$ 's to separate it from the next chunk of $B$ 's. The last chunk of $B$ 's must, of course, end with a $B$ . Thus our sequence must look like this :

\[\boxed{\quad A\text{'s} \quad}} \boxed{\quad B\text{'s} \quad}} BAA \boxed{\quad A\text{'s} \quad}} \cdots \boxed{\quad B\text{'s} \quad}}BAA \boxed{\quad A\text{'s} \quad}} \boxed{\quad B\text{'s} \quad}} B \boxed{\quad A\text{'s} \quad}} ,\] (Error compiling LaTeX. Unknown error_msg)

where each box holds an even number of letters (possibly zero).

If we want a sequence with $2k$ chunks of $B$ 's, then we have $(2n - 6k + 2)$ letters with which to fill the boxes. Since each box must have an even number of letters, we may put the letters in the boxes in pairs. Then we have $(n - 3k + 1)$ pairs of letters to put into $4k + 1$ boxes. By a classic balls-and-bins argument, the number of ways to do this is $\binom{n + k + 1}{4k} .$ It follows that the total number of desirable sequences is $\sum_k \binom{n + k + 1}{4k} .$ For $n = 7$ , this evaluates as $\binom{8}{0} + \binom{9}{4} + \binom{10}{8} = 1 + 126 + 45 = \boxed{172}$ .

@@ Line 46: / Line 46: @@
 If we want a sequence with <math>2k</math> chunks of <math>B</math>'s, then we have <math>(2n - 6k + 2)</math> letters with which to fill the boxes. Since each box must have an even number of letters, we may put the letters in the boxes in pairs. Then we have <math>(n - 3k + 1)</math> pairs of letters to put into <math>4k + 1</math> boxes. By a classic balls-and-bins argument, the number of ways to do this is
 <cmath>
-\binom{n + k + 1}{4k + 1} .
+\binom{n + k + 1}{4k} .
 </cmath>
 It follows that the total number of desirable sequences is
 <cmath>
-\sum_k \binom{n + k + 1}{4k + 1} .
+\sum_k \binom{n + k + 1}{4k} .
 </cmath>
 For <math>n = 7</math>, this evaluates as <math>\binom{8}{0} + \binom{9}{4} + \binom{10}{8} = 1 + 126 + 45 = \boxed{172}</math>.

2008 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
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