chaitanyabhansali wrote:Please solve this for me with a lucid step-by-step solution!
question: In how many different ways can 3 identical green shirts and 3 identical red shirts be distributed among 6 children such that each child receives a shirt?
The reason we divide by 3! twice is that the internal order within each group does not matter, so we need to discount duplicate arrangements.
When arranging 6 different colors, we're counting these arrangements as different:
R, B, P, G, W, Y
R, B, P, G, Y, W
R, B, P, Y, G, W
etc.
But if we transition to two "groups", i.e replace B and P with red shirts (R) and W and Y with Green shirts (G), then all three arrangements above are actually only a single arrangement:
R, R, R, G, G, G
R, R, R, G, G, G
R, R, R, G, G, G
For a group of 3 (green, white, yellow shirts, for example), there are 3! different ways of arranging the internal order of the members in the group, all of which are acutally a single option when moving to G, G, G. Therefore, divide the original 6! by 3! to discount the duplicates of the green group, and 3! to discount the duplicates of the red group.
If the same problem had 2 identical green shirts, 2 identical red shirts, and 2 identical blue shirts, then the number of ways of distributing the shirts to six kids would be?
