A little combinatorics exercise for those of you who still like to count: The 21 3k-powers-of-ten words (thousand, million, billion, trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion, undecillion, duodecillion, tredecillion, quattuordecillion, quindecillion, sexdecillion, septendecillion, octodecillion, novemdecillion, vigintillion) may be combined with the 999 positive-integers-less-than-1000 words (let's call them n-words) to express all numbers < 10^66 (U.S. notation). Any given power word may be used zero or one time, but every power word used once must be accompanied by a single n-word. In addition, we have the option of adding one extra single n-word. Finally, if no power words at all are used, then we must choose a single n-word. Let the length of an English number expression be the total number of letters in its written form; i.e., separators (spaces, commas) and conjunctives (hyphens, the word 'and') are ignored. The lengths of the above power words are: 8, 7, 7, 8, 11, 11, 10, 10, 9, 9, 9, 11, 12, 12, 17, 13, 12, 15, 13, 14, 12. Tallying these, we get lengths of 7-17 occurring with a frequency (respectively) of: 2, 2, 3, 2, 3, 4, 2, 1, 1, 0, 1. Tallying the n-words, we get lengths of 3-24 occurring with a frequency (respectively) of: 4, 3, 6, 6, 3, 13, 22, 27, 18, 6, 12, 21, 39, 45, 45, 66, 114, 177, 183, 126, 54, 9. So, for example, the longest number expression < 10^66 should include every power word (with a total of 230 letters) plus 22 instances of a 24-letter n-word: 22*24 + 230 = 758. How many 758-letter number expressions < 10^66 are there? 9^22. Correct? How many 42-letter number expressions < 10^66 are there?