Mock AIME 3 Pre 2005 Problems/Problem 5

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.

Mock AIME 3 Pre 2005 (Problems, Source)
Preceded by Problem 4	Followed by Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

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