Combinatorics with Twins
A team of 10 people were selected to fill up 4 positions: the King, the Bishop, the Knight and the Soldier. However, 2 of the 10 people are identical twins, and everyone is unable to tell them apart. If the twins are interchangable (regardless if they have been selected), how many ways can the positions be filled?
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1 solution
We may partition all the cases into three categories.
Neither of them is picked: In this case, we have four spots to fill with the eight remaining people and there are 8 × 7 × 6 × 5 = 1 6 8 0 ways of doing this.
Just one of them is picked: First, notice that it doesn't matter which one. So first we need to assign a position to the one that is picked. There are four positions available, and then we would have 3 empty positions to fill with the other eight people so there are 4 × 8 × 7 × 6 = 1 3 4 4 ways of doing this.
Both of them are picked: First, we have to fill two positions with them and as they are identical it doesn't matter who gets what so we simply choose 2 positions out of 4 and hand them to the two twins. Then we would have 2 empty spots to fill with the remaining 8 players and the total ways of doing this would be ( 2 4 ) × 8 × 7 = 3 3 6 .
Adding all the results we get 1 6 8 0 + 1 3 4 4 + 3 3 6 = 3 3 6 0 .