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Difference between revisions of "Mock AIME 3 Pre 2005 Problems/Problem 5"

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(→‎Solution 3)
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<cmath>O,C,C,O,C,C,O</cmath>
 
<cmath>O,C,C,O,C,C,O</cmath>
 
We want to incorporate <math>3</math> <math>C</math>s into the <math>4</math> intervals.
 
We want to incorporate <math>3</math> <math>C</math>s into the <math>4</math> intervals.
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If the <math>C</math>s are in <math>3</math> separate intervals, there are <math>4/choose3=4</math> possibilities.
+
 
−
If the <math>C</math>s are in <math>2</math> different intervals, there are $
+
If the <math>C</math>s are in <math>3</math> separate intervals, there are <math>{4\choose3}=4</math> possibilities.
 +
 
 +
If the <math>C</math>s are in <math>2</math> different intervals, then one has <math>2</math> <math>C</math>s and the other has <math>1</math> <math>C</math>. There are <math>2\cdot{4\choose2}=12</math> possibilities.
 +
 
 +
If the <math>C</math>s are in the same intervals, there are <math>4</math> possibilities.
 +
 
 +
In total, case <math>2</math> holds <math>(4+12+4)\cdot2^7=2560</math> possible words.
 +
 
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Case <math>3</math>: The word contains <math>2</math> <math>O</math>s.
  
 
==See Also==
 
==See Also==

Revision as of 09:17, 28 November 2019

Problem

In Zuminglish, all words consist only of the letters $M, O,$ and $P$. As in English, $O$ is said to be a vowel and $M$ and $P$ are consonants. A string of $M's, O's,$ and $P's$ is a word in Zuminglish if and only if between any two $O's$ there appear at least two consonants. Let $N$ denote the number of $10$-letter Zuminglish words. Determine the remainder obtained when $N$ is divided by $1000$.

Solution

Solution 1 (recursive)

Let $a_n$ denote the number of $n$-letter words ending in two constants (CC), $b_n$ denote the number of $n$-letter words ending in a constant followed by a vowel (CV), and let $c_n$ denote the number of $n$-letter words ending in a vowel followed by a constant (VC - the only other combination, two vowels, is impossible due to the problem statement). Then, note that:

  • We can only form a word of length $n+1$ with CC at the end by appending a constant ($M,P$) to the end of a word of length $n$ that ends in a constant. Thus, we have the recursion $a_{n+1} = 2(a_n + c_n)$, as there are two possible constants we can append.
  • We can only form a word of length $n+1$ with a CV by appending $O$ to the end of a word of length $n$ that ends with CC. This is because we cannot append a vowel to VC, otherwise we'd have two vowels within $2$ characters of each other. Thus, $b_{n+1} = a_n$.
  • We can only form a word of length $n+1$ with a VC by appending a constant to the end of a word of length $n$ that ends with CV. Thus, $c_{n+1} = 2b_n$.

Using those three recursive rules, and that $a_2 = 4, b_2 = 2, c_2=2$, we can make a table: \[\begin{array}{|r||r|r|r|} \hline &a_n&b_n&c_n \\ \hline 2 & 4 & 2 & 2 \\ 3 & 12 & 4 & 4 \\ 4 & 32 & 12 & 8 \\ 5 & 80 & 32 & 24 \\ 6 & 208 & 80 & 64 \\ 7 & 544 & 208 & 160 \\ 8 & 408 & 544 & 416 \\ 9 & 648 & 408 & 88 \\ 10 & 472 & 648 & 816 \\ \hline \end{array}\] For simplicity, we used $\mod 1000$. Thus, the answer is $a_{10} + b_{10} + c_{10} \equiv \boxed{936} \pmod{1000}$. (the real answer is $15936$.)

Solution 2 (combinatorics)

See solutions pdf.


Solution 3

Let $O$ denotes vowel and $C$ denotes consonants. Notice that there can be a maximum of 4 $O$s. Specifically, \[O,C,C,O,C,C,O,C,C,O\] Doing simple casework:

Case $1$: The word contains $4$ vowels.

For each $C$, there are $2$ choices. There is a total of $2^6=64$ possible words.

Case $2$: The word contains $3$ vowels.

Consider the word \[O,C,C,O,C,C,O\] We want to incorporate $3$ $C$s into the $4$ intervals.

If the $C$s are in $3$ separate intervals, there are ${4\choose3}=4$ possibilities.

If the $C$s are in $2$ different intervals, then one has $2$ $C$s and the other has $1$ $C$. There are $2\cdot{4\choose2}=12$ possibilities.

If the $C$s are in the same intervals, there are $4$ possibilities.

In total, case $2$ holds $(4+12+4)\cdot2^7=2560$ possible words.

Case $3$: The word contains $2$ $O$s.

See Also

Mock AIME 3 Pre 2005 (Problems, Source)
Preceded by
Problem 4 		Followed by
Problem 6
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