Probability of selecting letters in alphabetic order

[xcrunner448]

Homework Statement

There are 26 tiles in a bag, each with a different letter on it. You draw 5, without replacement. What is the probability that you pick the five letters in alphabetic order (e.g. C-H-R-T-W)

The Attempt at a Solution

The only way I could think of to do this was to find the total number of possible 5 letter sequences and the number of those sequences that are in alphabetic order, and then divide the two. There are ₂₆P₅=7893600 different sequences of letters. But I have not been able to think of a good way to calculate the number of those that are in alphabetic order. Any suggestions would be appreciated. Thank you!

 

[verty]

First try it with only 5 letters in the bag, then 6 letters. Try to see a pattern you can extend to 26 letters.

 

[Dick]

For every set of five letters, there is only one way to arrange it in alphabetic order. So the number of ways to choose five letters in alphabetic order is the same as the number of ways to choose five letters from 26.

 

[xcrunner448]

Ah, that makes sense. So for each possible combination of 5 letters there are 5!=120 possible arrangements and only 1 that is in alphabetic order, so the probability is 1/120. It's interesting that it doesn't even matter how many tiles are in the bag, only how many you are picking at a time. Thanks, Dick.