有些捷运（MRT）列车的车厢每节有 12 个座位。乘客每天乘坐列车在车站之间往返。 最靠近车门的座位被标记为专座，预留给弱势通勤者，即孕妇、老人、携带婴儿者或残障人士。如果车厢内有足够的座位满足所有弱势通勤者的需求，任何人都可以在专座上就坐。 在一次行程中，有 12 名乘客（其中 4 名是弱势通勤者）在首站登上了 B 车厢。

1. Determine the number of ways to seat the 4 vulnerable commuters in the 4 priority seats.

There are exactly $4$ priority seats in Car B. The $4$ vulnerable commuters must sit in these seats.
Number of ways to seat them = $4!$

2. Determine the number of ways to seat the remaining passengers and calculate the total number of arrangements.

There are $8$ remaining passengers to be seated in the $8$ ordinary seats.
Number of ways to seat them = $8!$

Total number of ways = $4! \times 8! = 24 \times 40320 = 967680$

1. Calculate the total number of ways to arrange all 12 passengers without any restrictions.

$\text{Total unrestricted ways} = 12!$

2. Calculate the number of ways to arrange the passengers such that the 2 particular girls are seated together, by treating them as a single grouped unit.

There are $12 - 2 + 1 = 11$ units to arrange (10 individual passengers + 1 group of 2 girls).
$\text{Ways the girls sit together} = 11! \times 2!$

3. Subtract the number of ways they sit together from the total unrestricted ways to find the number of arrangements where they are not beside each other.

$\text{Ways they do not sit together} = 12! - (11! \times 2!)$
$= 479\,001\,600 - 79\,833\,600$
$= 399\,168\,000$

1. Find the number of ways to select and arrange 4 out of the 5 girls to sit together.

Choose 4 girls to form a group and arrange them within the group:
$\text{Number of ways} = \binom{5}{4} \times 4! = 5 \times 24 = 120$

2. Find the number of ways to arrange the remaining 7 passengers.

There are $12 - 5 = 7$ other passengers.
$\text{Number of ways to arrange them} = 7! = 5040$

3. Use the gap method to place the group of 4 girls and the remaining 1 girl so they are separated.

The 7 other passengers create $8$ available spaces (including the ends of the row). To ensure exactly 4 girls are together, the group of 4 girls and the single remaining girl must be placed in different spaces.
$\text{Number of ways to assign spaces} = {}^{8}P_2 = 8 \times 7 = 56$

4. Calculate the total number of valid seating arrangements.

Multiply the number of ways from the independent choices:
$\text{Total ways} = 120 \times 5040 \times 56 = 33,868,800$