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A railway passes through four towns A, B, C, and D. The railway forms 08 Jun 2020, 09:46
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40% (02:05) correct 60% (02:30) wrong based on 70 sessions

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A railway passes through four towns A, B, C, and D. The railway forms a complete loop, as shown below, and trains go in both directions. Suppose that a trip between two adjacent towns costs one ticket. Using exactly eight tickets, how many distinct ways are there of traveling from town A and ending at town A? (Note that passing through 'A' somewhere in the middle of the trip is allowed.)
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A. 48
B. 64
C. 96
D. 128
E. 150
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D
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A railway passes through four towns A, B, C, and D. The railway forms 08 Jun 2020, 10:04
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nick1816
A railway passes through four towns A, B, C, and D. The railway forms a complete loop, as shown below, and trains go in both directions. Suppose that a trip between two adjacent towns costs one ticket. Using exactly eight tickets, how many distinct ways are there of traveling from town A and ending at town A? (Note that passing through AA somewhere in the middle of the trip is allowed.)
Attachment:
Untitled.png

A. 48
B. 64
C. 96
D. 128
E. 150

Given: A railway passes through four towns A, B, C, and D. The railway forms a complete loop, as shown below, and trains go in both directions. Suppose that a trip between two adjacent towns costs one ticket.
Asked: Using exactly eight tickets, how many distinct ways are there of traveling from town A and ending at town A?


Possible trips = A(D/B)(A/C)(D/B)(A/C)(D/B)(A/C)(D/B)A
A trip ADADABABA represents how 8 tickets are spent travelling.

The number of distinct ways of traveling from town A and ending at town A \(= 2^7 = 128\)

IMO D
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A railway passes through four towns A, B, C, and D. The railway forms 08 Jun 2020, 10:26
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Kinshook


Possible trips = A(D/B)(A/C)(D/B)(A/C)(D/B)(A/C)(D/B)A
A trip ADADABABA represents how 8 tickets are spent travelling.

The number of distinct ways of traveling from town A and ending at town A \(= 2^7 = 128\)

IMO D
Sir can you explain this step?
What PnC combination it uses, are the trip continuous? Are we allowed to move cross section? And Are all 8 tickets has to used at once?
Kindly revert.

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A railway passes through four towns A, B, C, and D. The railway forms 10 Jun 2020, 07:29
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From Point A one can move either to B or C

Let's say moving Forward (i.e A to B) : F
moving Backward (i.e A to C) : B


Keeping this in Mind, we have these possible arrangements:

FFFFFFFFF: 1 (example - ABCDABCDA)
BBBBBBBB: 1 (example - ADCBADCBA)
FFFFBBBB: 8!/(4!4!) = 70 (example - ABCDADCBA / ABABABCBA)
FFFFFFBB: 8!/(6!2!) = 28 (example - ABCDABCBA)
BBBBBBFF: 8!/(6!2!) = 28 (example - ADCBADCDA)

Total: 1+1+70+28+28 = 128 ways
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A railway passes through four towns A, B, C, and D. The railway forms 04 Jan 2025, 22:47
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