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 role and each box both are distinct.

Options

A. 150
B. 50
C. 60
D. 90

Answer

To solve the problem of packing 5 distinct rolls into 3 distinct boxes such that none of the boxes remain empty, we can use the principle of inclusion-exclusion and combinatorial counting. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have 5 distinct rolls and 3 distinct boxes. We need to pack the rolls into the boxes ensuring that no box is empty.

### Step 2: Use the Stirling Numbers of the Second Kind
The problem can be approached using Stirling numbers of the second kind, which count the ways to partition a set of n objects into k non-empty subsets. However, since our boxes are distinct, we will also need to multiply by k! (the number of ways to assign the subsets to the boxes).

### Step 3: Calculate the Stirling Number
The number of ways to distribute 5 distinct rolls into 3 non-empty boxes can be calculated using the formula:
\[ S(n, k) = \text{Stirling number of the second kind} \]
where \( n = 5 \) and \( k = 3 \).

The Stirling number \( S(5, 3) \) can be calculated as follows:
- \( S(5, 3) = 25 \)

### Step 4: Multiply by the Factorial of the Number of Boxes
Since the boxes are distinct, we multiply the Stirling number by \( 3! \) (the number of ways to assign the 3 groups to the 3 boxes):
\[ 3! = 6 \]

### Step 5: Calculate the Total Number of Ways
Now, we can calculate the total number of ways to pack the rolls:
\[ \text{Total ways} = S(5, 3) \times 3! = 25 \times 6 = 150 \]

### Conclusion
Thus, the total number of ways to pack the 5 distinct rolls into 3 distinct boxes such that none of the boxes remains empty is **150**.