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Sigma identity for a permutation? (Combinatorics)

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Hi everyone. I'm currently studying basic combinatorics in college and I came across a question I'm having some trouble with:

Part 1: k number of b***s are thrown into 10 boxes - seven green, three red. More than one ball can be placed in each box. The way the course writes this is D(10,k). I'm asked to write a summation identity for this problem in the following format:

Part 2: Test the identity for k = 3.

Can anyone help me out with this? I've tried writing several identities and none of them work for k = 3. I believe the number of combinations I'm supposed to get is 220 and have been unsuccessful in writing an identity that uses summation to calculate that number.

Thanks!

May 25, 2015

#2
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Is there a maximum value for k ?

What do the 7 green boxes and 3 red boxes have to do with it ?  That doesn't seem to affect the problem at all, that is, 10 boxes all of the same colour would appear to produce the same problem.

Getting the answer 220 for k=3 is easy enough and that seems to give some insight into how to proceed. However, it opens up questions such as, suppose, for example, k=9, then we could put 2 b***s in one box and the other 7 in 7 separate boxes, or we could have two boxes each containing 2 b***s with the other 5 in separate boxes, or three  boxes each containing two b***s or 4 boxes each containing 2 b***s or one or two or three boxes each containing 3 b***s or ..... , 1 or 2 boxes each containing 4 b***s, and so on. Is there a function, or does there exist a function that will cover all of those possibilities ? What sort of function have you been trying ? Would a piece of computer code (an algoritm) be allowable as a function ?

May 27, 2015

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I don't know how to do this problems BUT are the b***s identical or different from one another?

May 26, 2015
#2
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