# Math Help - Re-arranging letters in a word...

1. ## Re-arranging letters in a word...

This is my solution to the following question. Case 3, seems to be wrong though.

How many 4 letter arrangements of the word TOMORROW are there?

So i separate letters into their own "buckets": T, OOO, M, RR, W

So here are the cases:

Case1: 4 letters are all different : 5C4 * 4!
Case2: 2 Letters are same, 2 Letters are different: 2C1 * 4C2 * 4!/2!
Case3: 2 letters are same, 2 letters are same: 2C1 * 1C1 * 4!/2!
Case4: 3 Same, 1 Different: 1C1 * 4 C 1 * 4!/3!

Add them all up at the end and you get the answer.

But did i do Case3 right?

2. Hello, Saibot!

This is my solution to the following question. Case 3, seems to be wrong though.

How many 4-letter arrangements of the word TOMORROW are there?

So i separate letters into their own "buckets": T, OOO, M, RR, W

So here are the cases:

Case 1: 4 letters are all different: . $\left(_5C_4\right)(4!)$

Case 2: 2 Letters are same, 2 Letters are different: . $\left(_2C_1\right)\left(_4C_2\right)\left(\frac{4! }{2!}\right)$

Case 3: 2 letters are same, 2 letters are same: . $\left(_2C_1\right)\left(_1C_1\right)\left(\frac{4! }{2!}\right)$

Case 4: 3 Same, 1 Different: . $\left(_1C_1\right)\left(_4C_1\right)\left(\frac{4! }{3!}\right)$

Add them all up at the end and you get the answer.

But did i do Case 3 right?

Case 3: Two letters are same, two letters are same.

The four letters must be: $O,O,R,R.$

And there are: . $\frac{4!}{2!2!}$ arrangements.