Question 1153346: A school dance committee is made up of 3 freshmen, 6 sophomores, 3 juniors, and 2 seniors.
How many ways are there to sit the committee in a row at a meeting if the students must sit together by grade?
How many ways are there to sit the committee in a row at a meeting if the freshman, sophomores, and juniors must sit by grade, but the seniors can sit wherever they want?
Answer by jim_thompson5910(35256)   (Show Source):
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Problem 1
Question:
A school dance committee is made up of 3 freshmen, 6 sophomores, 3 juniors, and 2 seniors.
How many ways are there to sit the committee in a row at a meeting if the students must sit together by grade?
Answer: 1,244,160 (a little over 1 million)
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Work Shown:
The students must sit together by grade, so we effectively have 4 sections or blocks. There are 4! = 4*3*2*1 = 24 ways to arrange 4 blocks. Let A = 24.
Within the freshman block, there are 3! = 3*2*1 = 6 ways to arrange these students. Let B = 6.
Within the sophomores block, there are 6! = 6*5*4*3*2*1 = 720 ways to arrange these students. Let C = 720.
Within the juniors block, there are 3! = 3*2*1 = 6 ways to arrange these students. Let D = 6.
Within the seniors block, there are 2! = 2*1 = 2 ways to arrange these students. Let E = 2.
Put together, there are A*B*C*D*E = 24*6*720*6*2 = 1,244,160 different permutations possible such that the students must sit together by grade.
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Problem 2
Question:
A school dance committee is made up of 3 freshmen, 6 sophomores, 3 juniors, and 2 seniors.
How many ways are there to sit the committee in a row at a meeting if the freshman, sophomores, and juniors must sit by grade, but the seniors can sit wherever they want?
Answer: 3,110,400 (slightly over 3 million)
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Work Shown:
There are 3 blocks: freshmen, sophomores, juniors
Then there are 2 seniors who can sit wherever they want.
Effectively, there are 5 "people" (3 of which are groups of people)
A = 5! = 5*4*3*2*1 = 120 ways to arrange the five "people"
B = 3! = 3*2*1 = 6 ways to arrange the freshmen within any particular block
C = 6! = 6*5*4*3*2*1 = 720 ways to arrange the sophomores within any particular block
D = 3! = 3*2*1 = 6 ways to arrange the juniors within any particular block
A*B*C*D = 120*6*720*6 = 3,110,400 is the total number of permutations in which the freshmen, sophomores, and juniors must sit together but the seniors can sit anywhere they want.
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