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- Homework Statement:
- Decide in how many ways the letters of ABRACADABRA can be arranged in a row if C, D and R are not to be together

- Relevant Equations:
- 11!/5!/2!/2! - 9!*3!/5!/2!/2

Hi everyone

Can anyone help with this combinatorics problem? I have the answers, but don't understand how the answer was derived.

Here's my attempt and reasoning.

Step 1: Determine all possible combinations

Since A, B and R have multiple letters, the number of possible combinations is given by 11!/5!/2!/2!

Step 2: Find all possible ways the unwanted condition can occur.

Treat C,D,R as a single block. Letters can be thought of as 8 letters + a block of 3. Within the block, there are 3! ways to arrange C, D and R.

So 9!*3!/5!/2! = 9072.

Subtracting the unwanted combinations from the total possible combinations gives 83,160 - 9072 = 74, 088.

However, the answer is 78,624, which you get with 83,160 - (9072/2).

I knew the second R was significant, but I didn't know how. For example, you might get four-letter strings like RCDR, which could be read as either RCD or CDR, but I didn't know how to factor that into the calculation.

Can someone explain to me why 9072 needed to be halved?

Thanks

Can anyone help with this combinatorics problem? I have the answers, but don't understand how the answer was derived.

Here's my attempt and reasoning.

Step 1: Determine all possible combinations

Since A, B and R have multiple letters, the number of possible combinations is given by 11!/5!/2!/2!

Step 2: Find all possible ways the unwanted condition can occur.

Treat C,D,R as a single block. Letters can be thought of as 8 letters + a block of 3. Within the block, there are 3! ways to arrange C, D and R.

So 9!*3!/5!/2! = 9072.

Subtracting the unwanted combinations from the total possible combinations gives 83,160 - 9072 = 74, 088.

However, the answer is 78,624, which you get with 83,160 - (9072/2).

I knew the second R was significant, but I didn't know how. For example, you might get four-letter strings like RCDR, which could be read as either RCD or CDR, but I didn't know how to factor that into the calculation.

Can someone explain to me why 9072 needed to be halved?

Thanks

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