Question 1152119: A school dance committee is made up of 2 freshman, 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?
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20064)   (Show Source):
You can put this solution on YOUR website!
We seat the freshmen, sophomores and juniors like this:
ffssssssjjj
That's 2!∙6!∙3! ways
For each of those ways, there are two cases:
Case 1: the two seniors, say Ann and Bill, sit side by side:
_f_f_s_s_s_s_s_s_j_j_j_
There are 12 places they can sit together.
There are 2 ways they can sit, Ann on Bill's left and Ann on Bill's right.
That's 12∙2 ways for case 1.
Case 2: the two seniors, Ann and Bill, do not sit side by side:
There are 12 places Ann can sit. For each of the 12 places Ann can sit,
there remains 11 places Bill can sit.
That's 12∙11 ways for case 2.
Answer: 2!∙6!∙3!(12∙2 + 12∙11) = 1347840 ways.
Edwin
Answer by ikleyn(52903)  (Show Source):
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