combinations & permutations

[carebear]

How many different 5 letter arrangements can be made from the letters in the word: October?

Case 1: 2 o's C(5,3) x 5!/2! = 600

Case 2: 1 o C(6,5) x 5! = 720

Case 3: no o's C(5,5) x 5! = = 120

For a total of 1440 ways....is this correct?

 

[soroban]

Hello, carebear!

How many different-5 letter arrangements can be made from the letters in the word: OCTOBER?

\(\displaystyle \text{Case 1: two O's}\quad C(5,3)\times \frac{5!}{2!} \:=\: 600\)

\(\displaystyle \text{Case 2: one O}\quad C(6,5)\times 5! \:=\: 720\) . This is incorrect

\(\displaystyle \text{Case 3: no O's}\quad C(5,5) \times 5! \:=\:120\)

For a total of 1440 ways.
is this correct?

There are two O;s, and five "Others".


Case 2: one O and four Others.

We have one O.
From the 5 Others, select 4 more letters: .\(\displaystyle C(5,4)\) ways.
Then arrange the 5 letters in \(\displaystyle 5!\) ways.

\(\displaystyle \text{Answer: }\:C(5,4) \times 5! \:=\:600\text{ ways.}\)



Case 3 could have been reasoned like this:

No O's . . . take the five Others and arrange them: .\(\displaystyle P(5,5) = 120\text{ ways.}\)