Discussion
A substitution takes a ‘word’ written in the letters X and Y and gives a new word by replacing each (old) X with a Y, and each (old) Y with XY. For instance, it converts YXY to XYYXY. Starting with the 1-letter word \(w_{0}\) = X, we get new words \(w_{1}\) = Y, \(w_{2}\) = XY, \(w_{3}\) = YXY and so on with each \(w_{n+1}\) obtained from \(w_{n}\) by applying the substitution. If (l, m) is the pair with l = number of X in \(w_{10}\) and m = number of Y in \(w_{9}\). Then (l, m) = ............ .
Answer: B
The pair is (34, 34).
Each X becomes Y and each Y becomes XY. So if a word has x letters X and y letters Y, the next word has y letters X (one from each Y) and x + y letters Y (one from each X and one from each Y).
\[ \begin{aligned} x_{n+1} &= y_{n} \\ y_{n+1} &= x_{n} + y_{n} \end{aligned} \]
Starting from \(w_{0}\) = X, with (x, y) = (1, 0), the counts of Y run 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 for \(w_{0}\) to \(w_{10}\), the Fibonacci numbers. So m, the number of Y in \(w_{9}\), is 34, and l, the number of X in \(w_{10}\), equals the number of Y in \(w_{9}\), also 34.
(21, 34) and (34, 21) come from slipping one step in either count, and (55, 34) takes the Y count of \(w_{10}\) for l.
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