Passengers are to travel by a double decked bus which can accommodate 13 in the upper deck and 7 in the lower deck. The number of ways that they can be distributed if 5 refuse to sit in the upper deck and 8 refust to sit in the lower deck is

Options

A. 25
B. 21
C. 18
D. 15

Answer

To solve the problem of distributing passengers in a double-decked bus with specific seating restrictions, we can break it down into a series of logical steps.

### Step-by-Step Solution:

1. **Identify the Total Passengers and Restrictions**:
- Total seats in the upper deck = 13
- Total seats in the lower deck = 7
- Passengers refusing to sit in the upper deck = 5
- Passengers refusing to sit in the lower deck = 8

2. **Determine the Passengers Who Can Sit in Each Deck**:
- Since 5 passengers refuse to sit in the upper deck, the remaining passengers who can sit in the upper deck = Total passengers - 5.
- Since 8 passengers refuse to sit in the lower deck, the remaining passengers who can sit in the lower deck = Total passengers - 8.

3. **Calculate the Total Number of Passengers**:
- Let \( x \) be the total number of passengers.
- We know that the upper deck can accommodate 13 passengers and the lower deck can accommodate 7 passengers, hence:
\[ x = 13 + 7 = 20 \]

4. **Determine the Distribution of Passengers**:
- Passengers who can sit in the upper deck = \( x - 5 = 20 - 5 = 15 \)
- Passengers who can sit in the lower deck = \( x - 8 = 20 - 8 = 12 \)

5. **Calculate the Number of Ways to Choose Passengers for Each Deck**:
- We need to select 5 passengers from the 15 who can sit in the upper deck:
\[ \text{Ways to choose 5 for upper deck} = \binom{15}{5} \]
- The remaining passengers (after selecting for the upper deck) will be 10 (15 - 5). We need to select 7 passengers for the lower deck from these remaining passengers:
\[ \text{Ways to choose 7 for lower deck} = \binom{10}{7} \]

6. **Calculate the Combinations**:
- Using the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \):
\[ \binom{15}{5} = \frac{15!}{5! \cdot 10!} = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1} = 3003 \]
\[ \binom{10}{7} = \binom{10}{3} = \frac{10!}{3! \cdot 7!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]

7. **Calculate the Total Number of Ways**:
- The total number of ways to distribute the passengers is the product of the combinations calculated:
\[ \text{Total ways} = \binom{15}{5} \times \binom{10}{7} = 3003 \times 120 = 360360 \]

### Final Answer:
The total number of ways that the passengers can be distributed in the bus is **360360**.