Re: In how many ways can 3 prizes be distributed among 5 students when
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20 May 2021, 00:06
Let the 3 distinct prizes be: X —-Y—-Z
Let the 5 people be: A, B, C, D, and E
Since no one person can receive all 3 prizes, there are 2 ways we can distribute the prizes:
[1–1–1] or [2–1]
(Scenario 1)
Distribute the prizes so that 3 different people each get 1 prize
(1st)out of the 5 people, how many ways are there to choose the 3 people who will get the prize:
“5 choose 3” = 5! / 2! 3! = 10
And
(2nd)for each of the 10 possible groups, we can divide the 3 distinct prizes among 3 different people in:
3! = 6 ways
Scenario 1:
(10) * (6) = 60 ways
Scenario 2:
Distribute the prizes so that 1 person gets 2 of the distinct prizes and another 2nd person gets 1 of the prizes [2–1]
(1st) how many ways are there to choose the 2 people out of the 5 total who will receive the prizes?
“5 choose 2” = 5! / 2! 3! = 10 ways
And
(2nd) how many ways are there to divide the 3 prizes up such that there is one stack of 2 prizes and another stack of 1 prize
“3 choose 2” * “1 choose 1” = 3 ways
And
(3rd)how many ways are there to distribute the two distinct “stacks” of prizes to the 2 people chosen?
2! = 2 ways
Scenario 2:
(10) * (3) * (2) = 60 ways
Answer
60 + 60 =
120 ways
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