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February 25th, 2017, 09:37 PM   #1
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Combinations where only one element can repeat.

Hi!

Let's say I have the letters A, B, C, D. I want to know the number of combinations in the space _, _, _, or 3, where repetitions are allowed. For this, I can apply the formula $\displaystyle C=n ^ x$. In the example would be $\displaystyle C=4 ^ 3=64$.

My question is:
How do I calculate this, but if only one letter can be repeated? Let's just say "A" can be repeated.

For instance:
A, A, A
A, B, A
C, A, A
D, A, B (This one does not repeat any, but is part of the result)
...
...

Thanks!
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February 25th, 2017, 10:03 PM   #2
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Does order matter?

does $ABD=DAB$ ?

If not...

this will be the sum of the number of combinations with no repeats plus the number of combinations with 1 repeat.

There are $\binom{4}{3}=4 \text{ combinations with 0 repeats.}$

There are $\binom{4}{1}\binom{3}{1}= 12 \text{ combinations with 1 repeat.}$

$\text{Thus a total of 4+12=16 combinations allowing a single repeat.}$
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February 25th, 2017, 11:53 PM   #3
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Quote:
Originally Posted by romsek View Post
Does order matter?

does $ABD=DAB$ ?

If not...

this will be the sum of the number of combinations with no repeats plus the number of combinations with 1 repeat.

There are $\binom{4}{3}=4 \text{ combinations with 0 repeats.}$

There are $\binom{4}{1}\binom{3}{1}= 12 \text{ combinations with 1 repeat.}$

$\text{Thus a total of 4+12=16 combinations allowing a single repeat.}$
hrm from your formula I see that order does matter.

In this case all we have to do is just subtract off those combos where elements are repeated 3 times.

There will be 4 of these.

So there will be $4^3 - 4 = 60$ different combinations allowing a single repeat.
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