How many sums from combination of coins? One 2 pence and one 5 pence, gives 4 different sums, 0, 2, 5, 7. With two 2 pence and two 5 pence I can make 0, 2, 4, 5, 7, 9, 10, 12, 14. With $n$ different coins of $k$ many, how many sums can be made?

This depends on the value of the coins. If the values of the $n$ coins are all vastly different, then each of the $k+1$ choices ($0,1,\dots,k$) for the number of coins leads to $(k+1)^n$ many different sums. However, if the values of the coins are relatively close, then there can be overlap in the sums.

Your question seems related to the Froebenius coin problem.