How many distinct ordered arrangements can be made with the letters of the word TENNESSEE?

Question

How many distinct ordered arrangements can be made with the letters of the word TENNESSEE?

Answer 1

Imagine you have 9 slots, which you must fill with letters from the word TENNESSEE to create different arrangements:

 

_ _ _ _ _ _ _ _ _

 

First, you must place the letter T somewhere, and so you have 9 options for where to put it. But for the sake of a visual, let's imagine we put it in the front:

 

T _ _ _ _ _ _ _ _

 

Now we have to put the two N's somewhere. Well, we have 8 slots left, and we're choosing two of them for the N's. This literally equates to the combination 8C2 (said as "8 choose 2" or "8 combination 2"). Let's imagine we put them in the next two slots:

 

T N N _ _ _ _ _ _ 

 

Now we have to put the two S's somewhere. But now we only have 6 slots left, and we're choosing two of them for the S's. This gives us 6C2 ways of doing that. So let's put them down next to the N's:

 

T N N S S _ _ _ _

 

Finally, we have four E's to place, but now we only have 4 slots left to choose from. So there's only one way to place the E's once all of the other letters are placed:

 

So putting all of this together, we have:

 

(9 ways to place T)*(8C2 ways to place the N's)*(6C2 ways to place the S's)*(1 way to place the E's)

 

Which is:

 

9*(28)*(15)*(1) = 3,780