When we calculate the number of permutations of alike objects with the formula $$\frac{n!}{p! \cdot q! \cdot r!}$$ when $p$ objects are of one kind, $q$ objects are of one kind, and $r$ objects are of one kind and so on, what are we actually getting?
How is the number of permutations of alike objects different from from the normal number of permutations
combinatoricspermutations
Best Answer
If we set $5$ persons on a row then there are $5!=120$ possibilities for that.
If we have $5=r+g+b$ balls on a row from which $r=2$ are red, $g=1$ are green and $b=2$ are blue then at first hand there are $5!=120$ possibilities for that.
But if we can distinguish the balls only by color then we "see" less possibilities.
Results like $R_1G_1R_2B_1B_2$ and $R_2G_1R_1B_2B_1$ are look-alikes.
So if we are after distinguishable possibilities then we are overcounting, and result $RGRBB$ is counted exactly $2!1!2!$ times.
This shows that the overcounting can be repaired by dividing with factor $2!1!2!$ resulting in $\frac{5!}{2!1!2!}=30$ possibilities.
Of course this can easily be generalized.