How many 3 letter combos can be made from the word proportion?

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- Jan 30th 2007, 07:40 AMthefirsthokageLetter Combinations
How many 3 letter combos can be made from the word proportion?

- Jan 30th 2007, 09:42 AMThePerfectHacker
- Jan 30th 2007, 12:37 PMSoroban
Hello, thefirsthokage!

There is no neat formula for this one.

I made a list . . . the shortest possible (I think).

Quote:

How many three-letter combos can be made from the word PROPORTION?

3 letters the same: $\displaystyle OOO$ . . . one way.

2 letters the same: there $\displaystyle 3$ choices of the matching pair ($\displaystyle OO,\,PP,\text{ or }RR)$

. . and $\displaystyle 5$ choices for the third letter . . . $\displaystyle 3 \times 5 \:=\:15$ ways.

3 differerent letters: there are $\displaystyle \binom{6}{3} = 20$ ways.

Therefore, there are: $\displaystyle 1 + 15 + 20 \:=\:\boxed{36}$ three-letter combos.

- Jan 30th 2007, 12:58 PMPlato
Here is a second way the get the same answer as Soroban.

The coefficient of $\displaystyle x^3$ in the expansion of $\displaystyle \left( {\sum\limits_{k = 0}^3 {x^k } } \right)\left( {\sum\limits_{k = 0}^2 {x^k } } \right)^2 \left( {1 + x} \right)^3$ is 36.

However that is assuming that the word ‘combos’ means the same thing as multi-set. In the above reply <p,o,p> is a multi-set and is counted only once. If this word ‘combos’ means three letter strings the we would count pop, ppo and opp.

What is the intended meaning of ‘combos’? - Jan 30th 2007, 01:57 PMthefirsthokage
Thanks guys for all your help. I do alright in calculus, but I suck so bad in statistics. I just have to keep trying, I guess.

And by combos, I mean combinations. - Jan 30th 2007, 02:10 PMPlato