8. There are three identical red balls, three identical yellow balls and three identical green balls. In how many different ways can they be split into three groups of three balls each?
(1 mark)
現有三個一模一樣的紅球、三個一模一樣的黃球和三個一模一樣的綠球。有多少種不同的方法把球分成三組使得每組各有三個球？
8. 10
8. Let R, Y, G denote red, yellow and green ball respectively. Call a group 'pure' if all balls are of the same colour. We have the following cases:
- 3 pure groups - there is only 1 way of grouping
- 2 pure groups - this is not possible (if two groups are pure, so is the third)
- 1 pure group - there are 3 ways of grouping , corresponding to choosing one colour (say,
$\mathrm{R}$ ) out of three to form a pure group (the other two groups must be YYG and YGG)
- no pure group - either every group contains $\mathrm{R}$ (2 ways, \{RYG, RYG, RYG\} or \{RYG, RYY, RGG\}), or otherwise the grouping is $\{R R X, R X X, X X X\}$ where each $\mathrm{X}$ is either $\mathrm{Y}$ or $\mathrm{G}$; there are two choices for the first $\mathrm{X}$, then two choices for the ' $\mathrm{XX}$ ' in 'RXX' (e.g. after RRY we can choose RYG or RGG for the second group, but not RYY), giving a total of $2+2 \times 2=6$ ways in this case.
Hence the answer is $1+0+3+6=10$.