A double-decker bus carry `(u+e)` passengers, `u` in the upper deck and `e` in the lower deck. Find the number of ways in which the `u+e` passengers can be distributed in the two decks, if `r(lt=e)` particular passengers refuse to go in the upper deck and `s(lt=u)` refuse to sit in the lower deck.

Explanation

To solve the problem of distributing \( u + e \) passengers in a double-decker bus with specific restrictions, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Passengers and Restrictions**:
- We have \( u \) seats in the upper deck and \( e \) seats in the lower deck.
- There are \( r \) passengers who refuse to go in the upper deck (they must sit in the lower deck).
- There are \( s \) passengers who refuse to sit in the lower deck (they must sit in the upper deck).

2. **Determine Available Seats**:
- Since \( r \) passengers must sit in the lower deck, the number of available seats in the lower deck becomes \( e - r \).
- Since \( s \) passengers must sit in the upper deck, the number of available seats in the upper deck becomes \( u - s \).

3. **Calculate Total Available Seats**:
- The total number of seats available for distribution is:
\[ (u - s) + (e - r) = u + e - r - s \]

4. **Selecting Passengers for the Upper Deck**:
- We need to choose \( u - s \) passengers for the upper deck from the total \( u + e - r - s \) passengers. The number of ways to choose these passengers is given by the combination formula:
\[ \binom{u + e - r - s}{u - s} \]

5. **Selecting Passengers for the Lower Deck**:
- After selecting passengers for the upper deck, the remaining passengers will be seated in the lower deck. The number of ways to choose \( e - r \) passengers for the lower deck from the remaining passengers is:
\[ \binom{e - r}{e - r} \]
- Since we are choosing all remaining passengers for the lower deck, this simplifies to 1.

6. **Final Calculation**:
- The total number of ways to distribute the passengers is:
\[ \text{Total Ways} = \binom{u + e - r - s}{u - s} \times 1 = \binom{u + e - r - s}{u - s} \]

### Final Answer:
\[ \text{Total Ways} = \frac{(u + e - r - s)!}{(u - s)!(e - r)!} \]