# Letter Combinations

• Jan 30th 2007, 07:40 AM
thefirsthokage
Letter Combinations
How many 3 letter combos can be made from the word proportion?
• Jan 30th 2007, 09:42 AM
ThePerfectHacker
Quote:

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,
$\frac{10!}{2!2!3!1!1!1!}$
• Jan 30th 2007, 12:37 PM
Soroban
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?
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.

• Jan 30th 2007, 12:58 PM
Plato
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’?
• Jan 30th 2007, 01:57 PM
thefirsthokage
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 PM
Plato
Quote:

Originally Posted by thefirsthokage
And by combos, I mean combinations.

Well strictly speaking, combinations do not involve repetitions.