How many 3 letter combos can be made from the word proportion?
Hello, thefirsthokage!
There is no neat formula for this one.
I made a list . . . the shortest possible (I think).
We have these letters: .$\displaystyle \begin{Bmatrix}O\,O\,O \\ P\,P \\ R\,R\\ T \\ I \\ N\end{Bmatrix}$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.
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’?