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 all the rolls and all the boxes are identical.

Options

A. (a) 3
B. (b) 0
C. (c) 1
D. (d) 2

Answer

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

### Step 1: Understand the Problem
We need to distribute 5 identical rolls into 3 identical boxes with the condition that no box can be empty.

### Step 2: Initial Distribution
Since no box can be empty, we must first place at least one roll in each box. This means we will start by putting 1 roll in each of the 3 boxes.

### Step 3: Calculate Remaining Rolls
After placing 1 roll in each of the 3 boxes, we have used 3 rolls (1 in each box). Therefore, we have:
\[ 5 - 3 = 2 \]
rolls left to distribute.

### Step 4: Distribution of Remaining Rolls
Now we need to distribute the remaining 2 rolls into the 3 boxes. Since the boxes are identical, we can consider the different ways to distribute these 2 rolls.

### Step 5: Possible Distributions
The ways to distribute 2 identical rolls into 3 identical boxes can be represented as the partitions of the number 2:
1. All 2 rolls in one box: (2, 1, 1) - This means one box has 3 rolls, and the other two have 1 roll each.
2. One roll in one box and one roll in another box: (1, 1, 1) - This is not possible since we have already placed 1 roll in each box.

Thus, the only valid distribution of rolls is:
- (3, 1, 1)

### Step 6: Count the Distinct Arrangements
Since both the rolls and the boxes are identical, the arrangement (3, 1, 1) is the only distinct way to pack the rolls.

### Conclusion
Thus, the total number of ways to pack the rolls into the boxes is:
\[ \text{Total Ways} = 1 \]

### Final Answer
The number of ways of packing the rolls is **1**.

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