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16 May 2022, 07:19
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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. \(24!*6!*2^6\)
A. \(24!*6\)
B. \(24!*12\)
C. \(24!*2^6\)
D. \(25!*6!\)
E. \(24!*6!*2^6\)
16 May 2022, 07:19
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Official Solution:
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. \(24!*6!*2^6\)
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
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. \(24!*6!*2^6\)
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
18 Aug 2025, 07:29
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