[Math] Find the number of five-letter words that use letters from $\{A, B, C, \ldots, Z\}$ in which all letters are different and are in alphabetical order.

combinationscombinatoricsdiscrete mathematics

Find the number of five-letter words that use letters from $\{A, B, C, \ldots, Z\}$ in which all letters are different and are in alphabetical order.

Examples of valid words are "ghost" and "adgps".

For this question I cannot use combination numbers (i.e. $n$ choose $k$) and I am NOT trying to count them all out by hand. For this answer. I started with there being

  • 26 choices for the 1st letter

  • 25 choices for the 2nd letter

  • 24 choices for the 3rd letter

  • 23 choices for the 4th letter

  • 22 choices for the 5th letter

This solves the first half of the condition (all the letters need to be different), but it does not help me for the alphabetical order part.

Best Answer

We have $26$ choices for the first letter, $25$ for the second... $22$ for the fifth.

So we can make $26 * 25 * 24 * 23 * 22$ strings of $5$ different letters.

Each alphabetical word of $5$ letters can have its letters arranged in $120$ ways.

So we have $\displaystyle \frac{(26)(25)(24)(23)(22)}{120}=\boxed{65780 }$ words.

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