# Math Help - Letter Combinations

1. ## Letter Combinations

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

2. Originally Posted by thefirsthokage
How many 3 letter combos can be made from the word proportion?
Number of letters is 10.

And divide by repeats,
$\frac{10!}{2!2!3!1!1!1!}$

3. Hello, thefirsthokage!

There is no neat formula for this one.
I made a list . . . the shortest possible (I think).

How many three-letter combos can be made from the word PROPORTION?
We have these letters: . $\begin{Bmatrix}O\,O\,O \\ P\,P \\ R\,R\\ T \\ I \\ N\end{Bmatrix}$

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

2 letters the same: there $3$ choices of the matching pair ( $OO,\,PP,\text{ or }RR)$
. . and $5$ choices for the third letter . . . $3 \times 5 \:=\:15$ ways.

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

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

4. Here is a second way the get the same answer as Soroban.
The coefficient of $x^3$ in the expansion of $\left( {\sum\limits_{k = 0}^3 {x^k } } \right)\left( {\sum\limits_{k = 0}^2 {x^k } } \right)^2 \left( {1 + x} \right)^3$ 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’?

5. 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.

6. Originally Posted by thefirsthokage
And by combos, I mean combinations.
Well strictly speaking, combinations do not involve repetitions.