HMMT 二月 2002 · 冲刺赛 · 第 20 题
HMMT February 2002 — Guts Round — Problem 20
题目详情
英文原题
- [7] The Antarctican language has an alphabet of just 16 letters. Interestingly, everyword in the language has exactly 3 letters, and it is known that no word’s first letter equalsany word’s last letter (for instance, if the alphabet were { a, b } then aab and aaa could notboth be words in the language because a is the first letter of a word and the last letterof a word; in fact, just aaa alone couldn’t be in the language). Given this, determine the maximum possible number of words in the language.
解析
英文解析
- The Antarctican language has an alphabet of just 16 letters. Interestingly, everyword in the language has exactly 3 letters, and it is known that no word’s first letter equalsany word’s last letter (for instance, if the alphabet were { a, b } then aab and aaa could notboth be words in the language because a is the first letter of a word and the last letterof a word; in fact, just aaa alone couldn’t be in the language). Given this, determine the maximum possible number of words in the language.
Solution: 1024 Every letter can be the first letter of a word, or the last letter of aword, or possibly neither, but not both. If there are a different first letters and b differentlast letters, then we can form a · 16 · b different words (and the desired conditions will bemet). Given the constraints 0 ≤ a, b ; a + b ≤ 16, this product is maximized when a = b = 8,
giving the answer.