# $18$ mice were placed in $3$ groups, with all groups equally large. In how many ways can the mice be placed into $3$ groups?

In my textbook, the given answer is $$18!/(6!)^3$$.

But my teacher's answer is $$18!/(3!)(6!)^3$$. He solved like -

Please review the attached answer and let me know which one answer is correct? Thank in advance!

• Please use MathJax. Jul 20 '20 at 5:35

It depends if groups are distinguishable. For e.g. if you put 6 mouse each in 3 different color boxes then your textbook is correct, if you put them in same color boxes then your teacher is correct.

Take a smaller case say 4 mouse and 2 boxes and then think about. You might want to write down all the possibilities to give you clarity.

Ans clearly mentions in the way textbook has treated order in which groups are formed matter (i.e. they are distinguishable you can label them 1,2,3)

• If the 3 groups mentioned in the question are two experimental groups and one control group, then??
– user749241
Jul 20 '20 at 5:44
• @AmritKrishnaChowdhury As long as you can distinguish the groups (by any categorization), then your book's answer is correct. But if you have two indistinguishable experimental groups and one control group (which is different from the experimental) i.e. something like $A,A,C$, then the answer will change (you will divide by $2!$ instead of $3!$ in your teacher's solution). Jul 20 '20 at 5:47
• @AnuragA is correct Jul 20 '20 at 5:48