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Permutation and Combinations — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSPermutation and Combinations3 Marks
Question
18 mice were placed in two experimental groups and one control group, with all groups equally large. In how many ways can the mice be placed into three groups?
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Answer

It is given that 18 mice were placed equally in two experimental groups and one control group i.e., three groups.
Each group is of 6 mice.
So, number of ways forming three group each of size $ '6' = \ ^{18}\text{C}_6\ ^{12}\text{C}_6\ ^6\text{C}_6​​​​​​​=\frac{18!}{6!\ 6!\ 6!}$
But in tish counting the order in wgich three particular groups are formed is also counted.
Clearly, when we from group the order must not be counted.
Now three particular groups can occur in 3! ways when order is counted.
So, actual number of ways $=\frac{18!}{6!\ 6!\ 6!}$

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