So if I was just doing them separately there would be 21 ways or 7 choose 5 ways. Do i just multiply it then 21 * 21? Unsure how to put them together.
Distribute 5 red balls and 5 blue balls into three distinct boxes
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combinatorics
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0distinct or indistinguishable balls? – 2017-01-31
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0the balls are identical while the boxes are distinguishable – 2017-01-31
2 Answers
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If the balls are distinct then choose one from three boxes for each ball.
If the balls are indistinguishable (aside from colour) then count ways to arrange two partitions and five red objects. Do the same for five blue objects. Multiply.
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You correctly calculated the number of ways to put 5 red indistinguishable balls into 3 distinguishable boxes.
Same for 5 blue balls.
There is an obvious 1-to-1 relation between "combination of 5 blue and 5 red balls" and a pair ("combination of red balls", "combination of blue balls"). So 21 * 21 is the answer.