An election, with a twins twist!!!
A city council consists of 10 members, from whom will be selected 4 officers: President, Vice-President, Secretary, and Ombudsman. But two of the members of the council are identical twins, and no one can tell them apart! If the twins are "interchangeable", whether elected to an office or not, in how many ways can the positions be filled ?
The answer is 3360.
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1 solution
There is ( 2 8 ) = 2 8 ways to choose 4 people with both twins, as we just have to select 2 non-twin people. There is then 2 ! 4 ! = 1 2 ways to fill the four positions with them.
Similarly, there is ( 3 8 ) + ( 4 8 ) = 5 6 + 7 0 = 1 2 6 ways to choose 4 people with one or no twin (or ( 4 9 ) = 1 2 6 if we do it by taking 4 people out of 8 non-twins and 1 twin). There is then 4 ! = 2 4 ways to fill the four positions with them.
Answer is then 2 8 ⋅ 1 2 + 1 2 6 ⋅ 2 4 = 3 3 6 0 .