Lets start by putting:

1 coin in the 1st cup

2 coins in the 2nd cup

3 coins in the 3rd cup

..........

8 coins in the 8th cup

9 coins in the 9th cup

Since 1+2+3+4+...+9 = 45 and we have 44 coins means that we need to remove 1 coin but if we do that we will end up with 2 cups that have same number of coins. Thus it is not possible to put 44 coins in 10 cups without using the same number of coins twice.