How many different combinations of three prime numbers, not necessarily distinct, have a sum of 51?
The prime numbers less than or equal to 51 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31. We can try all possible combinations of three of these numbers to see which ones add up to 51. There are C(12,3)=220 possible combinations, but some of them add up to the same number. For example, the combinations (2, 17, 32) and (2, 32, 17) both add up to 51. There are 10 different sums that can be obtained by adding three prime numbers less than or equal to 51, and each sum can be obtained in several different ways. The table below shows the different sums and the number of ways to obtain each sum.
Sum Number of ways
2 + 29 + 20 1
2 + 31 + 18 1
2 + 37 + 12 1
3 + 11 + 37 1
5 + 13 + 29 1
7 + 17 + 21 2
11 + 13 + 27 1
17 + 19 + 15 2
23 + 29 + 7 1
There are a total of 10 different combinations of three prime numbers, not necessarily distinct, that have a sum of 51.
