Judges will select 5 finalists from the 7 contestants entered in a singing competition. The judges will then rank the

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Judges will select 5 finalists from the 7 contestants entered in a singing competition. The judges will then rank the contestants and award prizes to the 3 highest ranked contestants: a blue ribbon for first place, a red ribbon for second place, and a yellow ribbon for third place. How many different arrangements of prize-winners are possible?

A. 10
B. 21
C. 210
D. 420
E. 1,260

[spoiler]OA=C[/spoiler]

Source: Princeton Review

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Total contestants = 7
Finalist needed = 5
Highest rank = 3
Judges will select 5 out of 7 contestants and how to select 3 top ratings from the 5.
$$7C_5\cdot5C_3$$
$$7C_5=\frac{7!}{5!\left(7-5\right)!}=\frac{7\cdot6\cdot5!}{5!\cdot2\cdot1}=\frac{42}{2}=21$$
and
$$5C_3=\frac{5!}{3!\left(5-3\right)!}=\frac{5\cdot4\cdot3!}{3!\cdot2\cdot1}=\frac{20}{2}=10$$
$$Therefore,\ 7C_5\cdot5C_3=21\cdot10=210$$
Answer = Option C

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This question is just asking us how many ways can the three candidates be 'Arranged' from the 7 initial.
We can ignore the initial 5 selection completely. The arrangement is important here,

Choosing 3 out of 7 = 7C3
Arranging the three = 3! ways

Thus, (7! / 3! 4!) * 3 ! = 210

Answer C

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M7MBA wrote:
Thu Jul 30, 2020 11:21 am
Judges will select 5 finalists from the 7 contestants entered in a singing competition. The judges will then rank the contestants and award prizes to the 3 highest ranked contestants: a blue ribbon for first place, a red ribbon for second place, and a yellow ribbon for third place. How many different arrangements of prize-winners are possible?

A. 10
B. 21
C. 210
D. 420
E. 1,260

[spoiler]OA=C[/spoiler]

Solution:

Since the finalists that do not make it to the top 3 ranks do not matter, we can ignore the selection of five finalists and calculate the number of ways to choose and order 3 contestants from a total of 7, which is 7P3 = 7!/(7 - 3)! = 7 x 6 x 5 = 210.

Answer: C

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