How many ways are there for 3 TYPES of medals? (no ties)
Any of the 6 runners can finish 1st
Any of the remaining 5 runners can finish 2nd
Any of the remaining 4 runners can finish 3rd
6 * 5 * 4 = = 120
How many ways are there for 3 TYPES of medals? (ties possible)
6 gold -----> C(6,6) = 1
5 gold -----> C(6,5) = 6
4 gold -----> C(6,4) = 15
3 gold -----> C(6,3) = 20
2 gold, 4 bronze ----> C(6,2) * C(4,4) = 15
2 gold, 3 bronze ----> C(6,2) * C(4,3) = 60
2 gold, 2 bronze ----> C(6,2) * C(4,2) = 90
2 gold, 1 bronze ----> C(6,2) * C(4,1) = 60
1 gold, 5 silver ----> C(6,1) * C(5,5) = 6
1 gold, 4 silver ----> C(6,1) * C(5,4) = 30
1 gold, 3 silver ----> C(6,1) * C(5,3) = 60
1 gold, 2 silver ----> C(6,1) * C(5,2) = 60
1 gold, 1 silver, 4 bronze ----> C(6,1) * C(5,1) * C(4,4) = 30
1 gold, 1 silver, 3 bronze ----> C(6,1) * C(5,1) * C(4,3) = 120
1 gold, 1 silver, 2 bronze ----> C(6,1) * C(5,1) * C(4,2) = 180
1 gold, 1 silver, 1 bronze ----> C(6,1) * C(5,1) * C(4,1) = 120
1 + 6 + 15 + 20 + 15 + 60 + 90 + 60 + 6 + 30 + 60 + 60 + 30 + 120 + 180 + 120 = 873