There are 3 colours of socks; blue, black and white and each colour has two pairs of socks and we need to distribute them to six people and the left leg socks are distinguishable from right leg socks. What are the possible ways of distribution?
Approach: align the right leg socks in a line in a random manner, the number of ways of distributing the left leg socks to this one particular distribution will be:
$$\frac{6!}{(2!)^3}$$
But this is the one way of distribution, What are the other possible ways?
Does squaring of the expression makes any sense, where order matters?
Best Answer
Since there are six different socks, they can be arranged in $6!$ orders.
Choose two of the six positions for the black socks, two of the remaining positions for the blue socks, and fill the final two positions with white socks, which can be done in $$\binom{6}{2}\binom{4}{2}\binom{2}{2} = \frac{6!}{4!2!} \cdot \frac{4!}{2!2!} \cdot \frac{2!}{2!0!} = \frac{6!}{2!2!2!}$$ distinguishable ways. This is what you counted.
There are three pairs of socks, which are distinguishable by their colors. Hence, there are $3!$ possible arrangements.
There are three pairs of socks, which are distinguishable by their colors. There are $3!$ ways to arrange the pairs. Each pair can be arranged in $2!$ ways. Hence, there are $3!(2!)^3$ possible arrangements.