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27 Jan 2022, 03:05
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C
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A class of 30 students take up a full row at a movie theater. If there are six couples among the students and if each couple wants to sit together, in how many different ways can the students sit?
A. \(24!*6\)
B. \(24!*12\)
C. \(24!*2^6\)
D. \(25!*6!\)
E. \(24!*6!*2^6\)
M37-106
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A. \(24!*6\)
B. \(24!*12\)
C. \(24!*2^6\)
D. \(25!*6!\)
E. \(24!*6!*2^6\)
M37-106
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04 Feb 2022, 03:15
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Bunuel
Number the students from 1 through 30.
Glue couples together: {1, 2}, {3, 4}, {5, 6}, {7, 8}, {9, 10}, {11, 12}.
We'd be left with 30 - 12 = 18 single students and together with couples' units we'd get 6 + 18 = 24 units: {1, 2}, {3, 4}, {5, 6}, {7, 8}, {9, 10}, {11, 12}, {13}, {14}, {15}, ..., {30}.
These 24 units can be arranged in 24! ways. Each couple within their unit can be additionally arranged in 2! ways (for example, {1, 2} or {2, 1}).
Therefore the final answer is \(24!*(2*2*2*2*2*2)= 24!*2^6\).
Answer: C.
General Discussion
27 Jan 2022, 03:22
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Bunuel
Suppose couples are single people. Then number of ways in which they will sit is 24!
Now each couple can sit in 2 way. There are six couples. 2^6
Total no of ways = 24!x2^6
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