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There are six runners in the 100-yard dash. How many ways are there for three medals to be awarded if ties are possible? (The runner or runners who finish with the fastest time receive gold medals, the runner or runners who finish with exactly one runner ahead receive silver medals, and the runner or runners who finish with exactly two runners ahead receive bronze medals.)
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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
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