A mathematical prodigy wishes to put 2 of his indistinguishable IMO gold medals and 2 of his indistinguishable IPhO gold medals in one row. How many distinct arrangements are possible?
This is a combinatorics problem, so I need to count the number of ways to choose the positions of the medals in the row. One way to approach this is to think of the row as having 4 slots, and I need to choose 2 of them for the IMO medals. The remaining 2 slots will automatically be filled by the IPhO medals. The number of ways to choose 2 slots out of 4 is given by the binomial coefficient (4 choose 2), which is equal to 4!/(2!2!) = 6. Alternatively, I can think of the row as having 4 slots, and I need to choose 2 of them for the IPhO medals. The remaining 2 slots will automatically be filled by the IMO medals. The number of ways to choose 2 slots out of 4 is again given by the binomial coefficient (4 choose 2), which is also equal to 6. Either way, I get the same answer, so I can be confident that I did not double-count or miss any cases. Therefore, the number of distinct arrangements is 6. 