2023 AMC 8 Problems/Problem 21
- of ways to split (1,2 , 3, 4, 5, 6, 7, 8, 9) into 3 equal groups
# of ways to split (1,2 , 3, 4, 5, 6, 7, 8, 9) into 3 equal groups
First we need to find the sum of each group when split. This is the total sum of all the elements divided by the # of groups. 
. Then dividing by 
we have 
so each group of must have a sum of 15. To make the counting easier we we will just see the possible groups 9 can be with. The posible groups 9 can be with with 2 distinct numbers are 
and 
. Going down each of these avenues we will repeat the same process for 
using the remaining elements in the list. Where there is only 1 set of elements getting the sum of 
, needs in both cases. After is decided the remaining 3 elements are forced in a group. Yielding us an answer of 
as our sets are 
and 
