Combinations removing duplications and repetition

Is there a general formula which can tackle the following:

given **n** objects and **r** spots, I want to calculate the number of possible combinations accounting for duplicates and repetition.

eg. n = 4, r = 3 (AAAB)

possible combinations:

AAA

AAB

result is 2

eg. n = 4, r = 3 (AABC)

possible combinations:

AAB

AAC

ABC

result is 3

eg. n = 5, r = 3 (AABBC)

possible combinations:

AAB

AAC

ABB

ABC

BBC

result is 5

Thank you

Re: Combinations removing duplications and repetition

Quote:

Originally Posted by

**s4276560** Is there a general formula which can tackle the following:

given **n** objects and **r** spots, I want to calculate the number of possible combinations accounting for duplicates and repetition.

eg. n = 4, r = 3 (AAAB)

possible combinations:

AAA

AAB

result is 2

eg. n = 4, r = 3 (AABC)

possible combinations:

AAB

AAC

ABC

result is 3

eg. n = 5, r = 3 (AABBC)

possible combinations:

AAB

AAC

ABB

ABC

BBC

result is 5

I think you are expecting a simple formula for this but there is none.

For one, we need to know not only how different kinds we have but also how many of each kind.

Here is an example.

$\displaystyle \{A,A,A,B,B,C\},~n=6<~\&~r=3$.

Expand $\displaystyle (1+x)(1+x+x^2)(1+x+x^2+x^3)=1+3x+5x^2+6x^3+5x^4+3x ^5+x^6$.

Looking at the coefficients we can answer your question.

$\displaystyle 6x^3$ tells us there are six possible combinations of three.

$\displaystyle 5x^4$ tells us there are five possible combinations of four, etc.

Whole courses have been given on counting problems.