# Task № 5.1

## Condition:

In a certain language $\mathrm{N}$, letters denote only 10 consonants and 5 vowels. Syllables in this language are of two types: either "consonant + vowel" or "consonant + vowel + consonant". A word in language $\mathrm{N}$ is any sequence of letters that can be broken down into syllables in an allowable manner. For example, the Russian word "кошка" (cat) could be a word in language $\mathrm{N}$, as it can be broken down into syllables as "кош-ка" (ko-sh-ka), whereas the word "гроза" (storm) could not exist in language $\mathrm{N}$ because it starts with two consonants that cannot be separated into syllables. How many eight-letter words are there in language $\mathrm{N}$?
Answer: 43750000

Exact match of the answer -1 point

## Solution.

Note that the number of vowels cannot be more than half of the total number of letters in the word, that is, more than 4, since in each syllable with a vowel there is at least one consonant. Also, the number of vowels cannot be less than $1 / 3$ of the number of letters in the word, since the number of consonants is at most twice the number of vowels. This means that 2 or fewer vowels in an eight-letter word is not possible. This means that the possible number of vowels is 4 or 3. In the first case, there is only one possible syllable division: cg-cg-cg-cg, and in the second case, there are three possible divisions: cgc-cgc-cg, cgcsc-cgc, cg-cgc-cgc (c - consonant, g - vowel).

Let's calculate the number of words for each of the four syllable divisions:

cg-cg-cg-cg: $10 * 5 * 10 * 5 * 10 * 5 * 10 * 5=10^{4} * 5^{4}$

cgc-cgc-cg: $10 * 5 * 10 * 10 * 5 * 10 * 10 * 5=10^{5} * 5^{3}$

cgc-cg-cgc: $10 * 5 * 10 * 10 * 5 * 10 * 5 * 10=10^{5} * 5^{3}$

cg-cgc-cgc: $10 * 5 * 10 * 5 * 10 * 10 * 5 * 10=10^{5} * 5^{3}$

The total number of eight-letter words in the language $\mathrm{N}$ is the sum of these four numbers, that is, $10^{4} * 5^{4}+3 * 10^{5} * 5^{3}=43750000$.