how many different strings can be made from the letters on orono using some or all of its letters
I know how to find the strings with all letters : 5!/3! but how do I find out the stings of different lengths?
As you say, length 5:
Length 4 can contain 3 or 2 o's.
- With 3 o's + 1 other, there is a choice 2 for the other letter, so the number is .
- With 2 o's both the other letters must be used, so that's .
- 3 o's:
- 2 o's + 1 other:
- 1 o + 2 others:
- 2 o's:
- 1 or 0 o's:
So I reckon that's .
That's a nice round number so I don't know if anyone knows of a quicker method?
I don't have a real shortcut, but I can't resist pointing out that the "easy way" if you are familiar with generating functions is to find the exponential generating function.
I.e., let be the number of words of length r and define . Then it's easy (if you know how) to see that
which on expansion yields
From this you can read off the number of words of length 1-5: 3, 7, 13, 20, 20.
This isn't much help for someone who doesn't know anything about generating functions, I guess, except it may give an incentive to learn.
Personally, I think generating functions are neat-- I guess you can tell that.