Solve the following problems independently of each other
Rollmall is a fast food outlet, which delivers the veg and non-veg rolls in its vicinity. Bawarchi is an employee, who is assigned to pack the rolls in the boxes for delivery. He has 5 rolls in hand to pack in the 3 boxes so that none of the boxes remains empty.
Find the number of ways of packing the rolls, if each roll is identical but each box is distinct.

Options

A. 3
B. 5
C. 6
D. 4

Answer

To solve the problem of packing 5 identical rolls into 3 distinct boxes such that none of the boxes remains empty, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have 5 identical rolls and 3 distinct boxes. The requirement is that each box must contain at least one roll.

2. **Initial Distribution**: Since each box must contain at least one roll, we can start by placing one roll in each of the 3 boxes. This ensures that no box is empty.

- After placing 1 roll in each of the 3 boxes, we have used 3 rolls.
- Remaining rolls = 5 - 3 = 2 rolls.

3. **Distributing Remaining Rolls**: Now we need to distribute the remaining 2 rolls into the 3 boxes. Since the rolls are identical, we can use the "stars and bars" theorem to find the number of ways to distribute these rolls.

4. **Applying the Stars and Bars Theorem**: The formula for distributing \( n \) identical items (rolls) into \( r \) distinct groups (boxes) is given by:

\[ \text{Number of ways} = \binom{n + r - 1}{r - 1} \]

Here, \( n \) is the number of remaining rolls (2), and \( r \) is the number of boxes (3).

5. **Substituting Values**: We substitute \( n = 2 \) and \( r = 3 \) into the formula:

\[ \text{Number of ways} = \binom{2 + 3 - 1}{3 - 1} = \binom{4}{2} \]

6. **Calculating the Binomial Coefficient**: Now we calculate \( \binom{4}{2} \):

\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

7. **Conclusion**: Therefore, the number of ways to pack the rolls into the boxes such that none of the boxes is empty is **6**.

### Final Answer:
The number of ways of packing the rolls is **6**.

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