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Six children, Arya, Betsy, Chen, Daniel, Emily, and Franco, are to be
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04 Oct 2021, 04:46
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Six children, Arya, Betsy, Chen, Daniel, Emily, and Franco, are to be seated in a single row of six chairs. If Betsy cannot sit next to Emily, how many different arrangements of the six children are possible?
Re: Six children, Arya, Betsy, Chen, Daniel, Emily, and Franco, are to be
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04 Oct 2021, 05:29
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Carcass wrote:
Six children, Arya, Betsy, Chen, Daniel, Emily, and Franco, are to be seated in a single row of six chairs. If Betsy cannot sit next to Emily, how many different arrangements of the six children are possible?
A. 240 B. 480 C. 540 D. 720 E. 840
Take the task of arranging the 6 students in a row and break it into stages. NOTE: We're going to ignore the chairs and just examine how many ways there are to arrange the 6 children in a row. The answer will be the same.
Stage 1: Arrange Arya, Chen, Daniel and Franco is a row We can arrange n unique "objects" in a row in n! ways So, we can arrange these 4 children in 4! ways (24 ways) So, we can complete stage 1 in 24 ways
IMPORTANT STEP: For each of the 24 arrangements of Arya, Chen, Daniel and Franco, add a space on either side of each child. For example, one of the possible arrangements is Chen, Daniel, Franco, Arya So, add spaces as follows: ___Chen___Daniel___Franco___Arya___ We'll now place Betsy in a space and place Emily in a different space. This will ENSURE that Betsy and Emily are not seated together.
Stage 2: Choose a space to place Betsy There are 5 spaces to choose from. So we can complete stage 2 in 5 ways
Stage 3: Choose a space to place Emily Once we select a space in stage 2, there are 4 spaces remaining to choose from. So we can complete stage 3 in 4 ways
By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus arrange all 6 children) in (24)(5)(4) ways (= 480 ways)
Answer: B
Note: the FCP can be used to solve the MAJORITY of counting questions on the GRE. So, be sure to learn it.