How many ways are there putting two red balls, two black balls, and six white balls into five different boxes by two in each box?
Same color balls cannot not be distinguished.
Answer
180 ways
Solution
① Put two red balls into the same box and put two black balls into different boxes.
There are 5 ways to choose the box into which two red balls are put.
As for choosing two boxes from four for two black balls, there are 4 × 3/2 × 1 = 6 ways. Therefore it's 5 × 6 = 30 ways altogether.
② Put two black balls into the same box and put two red balls into different boxes.
It's same as ① and it is 30 ways.
③ Put both of two red balls and two black balls are put into the same box, respectively.
Since there are five boxes to put two red balls and four boxes to put two black balls is 4, it's 5 × 4 = 20 ways altogether.
④ Put both of two red balls and two black balls into different boxes respectively.
Since there are 5 × 4/2 × 1 = 10 ways to choose boxes to put two red balls and there are also 10 ways to put two black balls, there are 10 × 10 = 100 ways altogether.
According to ① ~ ④, there are 30 + 30 + 20 + 100 = 180 ways in all.
Same color balls cannot not be distinguished.
Answer
180 ways
Solution
① Put two red balls into the same box and put two black balls into different boxes.
There are 5 ways to choose the box into which two red balls are put.
As for choosing two boxes from four for two black balls, there are 4 × 3/2 × 1 = 6 ways. Therefore it's 5 × 6 = 30 ways altogether.
② Put two black balls into the same box and put two red balls into different boxes.
It's same as ① and it is 30 ways.
③ Put both of two red balls and two black balls are put into the same box, respectively.
Since there are five boxes to put two red balls and four boxes to put two black balls is 4, it's 5 × 4 = 20 ways altogether.
④ Put both of two red balls and two black balls into different boxes respectively.
Since there are 5 × 4/2 × 1 = 10 ways to choose boxes to put two red balls and there are also 10 ways to put two black balls, there are 10 × 10 = 100 ways altogether.
According to ① ~ ④, there are 30 + 30 + 20 + 100 = 180 ways in all.
