Linear Arrangement with Constraints

Count the number of ways to arrange distinct or repeated objects in a row (line) subject to adjacency, ordering, or positional constraints.

tmua None Q13 View
13. Five runners competed in a race: Fred, George, Hermione, Lavender, and Ron.
Fred beat George. Hermione beat Lavender. Lavender beat George. Ron beat George.
Assuming there were no ties, how many possible finishing orders could there have been, given only this information?
A 1
B 6
C 12
D 18
E 24 F 120
turkey-yks 2014 Q31 View
Three domestic automobiles of brands A, B, and C and three foreign automobiles of brands X, Y, and Z will be displayed in a single row at an exhibition according to the following conditions.
  • Domestic and foreign automobiles will be arranged consecutively within their own groups.
  • Brand A automobile will be in the first or last position among all automobiles.
  • Brand X automobile will be in the first or last position among foreign automobiles.

Given this, in how many different ways can the automobiles be displayed?
A) 10
B) 12
C) 14
D) 16
E) 18
turkey-yks 2020 Q28 View
Two students from each of three different schools will participate in a chess tournament. In the first round of the tournament, each student will be paired with a student who is not from their own school.
Accordingly, in how many different ways can the pairing in the first round be done?
A) 6
B) 8
C) 9
D) 12
E) 15
turkey-yks 2023 Q27 View
In a textile workshop, cardboard boxes arranged in a row will pass sequentially through two machines that perform labeling.
The first machine writes NORMAL CUT on the first two cardboards, NARROW CUT on the next two cardboards, and continues in this way, writing NORMAL CUT on two cardboards and NARROW CUT on two cardboards in sequence, labeling the cardboards and sending them to the second machine. The second machine labels the first cardboard with S, the second cardboard with M, the third cardboard with L, and continues in this way, labeling the cardboards sequentially with S, M, and L.
Accordingly, what is the label of the 175th cardboard coming out of the second machine?
A) NORMAL CUT M B) NARROW CUT M C) NORMAL CUT S D) NARROW CUT L E) NORMAL CUT L
turkey-yks 2025 Q15 View
Sena has chosen a series A consisting of three films and a series B consisting of three films to watch a different film each of six specific evenings. For each series, Sena will not watch the second film without watching the first film of that series, and will not watch the third film without watching the second film of that series.
Accordingly, in how many different ways can Sena determine the order in which she will watch these six films?
A) 20 B) 24 C) 27 D) 30 E) 32