How many ways are there to order the letters ‘AABBC’? (For example, ‘ACABB’ and ‘AABBC’ are two ways.)

A. 5

B. 120

C. 60

D. 116

E. 30

How many ways are there to order the letters ‘AABBC’? (For example, ‘ACABB’ and ‘AABBC’ are two ways.)

A. 5

B. 120

C. 60

D. 116

E. 30

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Firstly, we will take the letters and distinguish same letters with subscripts, like A_1A_2B_1B_2C.

That gives 2+2+1=5 different symbols that can be permuted in 5! ways.

If we erase the subindices, this means e.g. A_1A_2, are now the same. This means that to get the total ways we need to divide 5! with 2!2!1!, so the total is:

\frac{5!}{2!2!1!}=\frac{5\cdot4\cdot3\cdot2\cdot1}{2\cdot1\cdot2\cdot1\cdot1}= 5\cdot3\cdot2=5\cdot6 =30.

So the correct answer is **(E)** 30.

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