Permutations & Combinations: Airplane seating; CCA — DHS 2025 H2 Math Prelim Paper 2
What this question tests
Question
Twelve students from three CCAs organised an overseas learning trip to Shanghai. It comprises five students from Tennis, four students from Bowling and three students from Softball. The available seats for the flight for them to choose from are shown in the diagram as follows:
- In how many different ways can the students be seated if there is a particular student who must be seated nearest to the emergency exit?
- Find the number of ways they can be seated if two particular students do not want to be seated on any of the aisle seats.
- Find the number of ways they can be seated if students of the same CCA must be seated together either front and back or left and right, and cannot be separated by an aisle. For example, the three students from Softball can be seated at Row 56 Seat G, H and Row 57 Seat G, but not Row 56 Seat H, I and Row 57 Seat G.
Show full worked solution▾
(a)
The particular student must sit in one of the 2 seats nearest to the emergency exit (Row 55 Seat I or Row 56 Seat I – both are in the same position; from the diagram there is 1 nearest seat at Row 55 I).
There are 13 remaining seats for the other 11 students to choose from.
(b)
There are 4 aisle seats (2 per aisle \(\times\) 2 aisles, from Rows 55–56 on each side). The 2 particular students must avoid all aisle seats, so they choose from the 7 non-aisle seats.
Stage 1 (seat the 2 restricted students in non-aisle seats): \(\dbinom{7}{2} \times 2! = 42\)
Stage 2 (arrange remaining 10 students in the remaining 12 seats): \(\dbinom{12}{10} \times 10! = \dfrac{12!}{2!}\)
(c)
The 5 Tennis students must occupy the middle segment (Columns D, E, F – 6 seats available across 3 rows). They arrange in columns E and F (2 seats per row): \(\binom{6}{5} \times 5! = 720\).
The 4 Bowling and 3 Softball students occupy Columns G, H, I (right segment).
Number of ways to arrange 4 Bowling + 3 Softball in 9 seats (3 rows \(\times\) 3 seats, front-back or left-right together without aisle separation): \(5 \times 4! \times 3! = 720\).
