Kenneth Rosen Ch 6 Counting

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#discrete

6 runners are in a 100 yard dash. Find the ways possible for 3 medals to be awarded if ties are possible.

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Here ties are possible and runner position depends on how many runner ahead of him finish the race and also one medal can be awarded to more than one person if there is a tie.

•  when there is one person for 1st position ,one for 2nd position and more than one for 3rd position.                              : 6c1 * 5c1 * (4c1 + 4c2 + 4c3 + 4c4 ) = 450
•  when there is one person for 1st position and more than one person for 2nd position in that case there is no person at 3rd position (bcoz already 3 are ahead of them) : 6c1 * ( 5c2 + 5c3 + 5c4 +5c5 ) = 156
• if there is two person for 1st position then there is no person for 2nd position and at 3rd position can be more than  one person : 6c2 * ( 4c1 +4c2 +4c3 +4c4 ) = 225
• if there is more than two person for 1st position then there is no one at place 2nd or 3rd  : ( 6c3 + 6c4 + 6c5 +6c6 ) =42   summing all of them is equal to  873
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why it is not possible that the first medal is given to 3 people, second one to 2 persons, and remaining can get the third medal ??
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if there is 3 people at first position then no one can be second or third bcoz already 3 person are ahead of them. now the next person is at position 4 ( position is decided by how many runner ahead of him finish the race ).

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