Out of 3 books on Economics, 4 books on Corporate Strategy and 5 books on Philosophy
what is the number of different collections of books if atleast one book on each subject is taken ?

Options

A. 2365
B. 3255
C. 4224
D. 1236

Answer

To solve the problem of finding the number of different collections of books if at least one book on each subject is taken, we can follow these steps:

### Step 1: Identify the total number of books in each category
- Economics: 3 books
- Corporate Strategy: 4 books
- Philosophy: 5 books

### Step 2: Calculate the number of ways to choose books from each category
For each category, we can use the formula for the number of ways to choose books, which is given by \(2^n - 1\), where \(n\) is the number of books in that category. This formula accounts for all possible selections of books (including choosing none) and subtracts the case where no book is chosen.

1. **Economics**:
- Number of ways to choose at least one book = \(2^3 - 1 = 8 - 1 = 7\)

2. **Corporate Strategy**:
- Number of ways to choose at least one book = \(2^4 - 1 = 16 - 1 = 15\)

3. **Philosophy**:
- Number of ways to choose at least one book = \(2^5 - 1 = 32 - 1 = 31\)

### Step 3: Calculate the total number of different collections
To find the total number of different collections of books, we need to multiply the number of ways to choose books from each category:

\[ \text{Total collections} = (\text{Ways from Economics}) \times (\text{Ways from Corporate Strategy}) \times (\text{Ways from Philosophy}) \]

Substituting the values we calculated:

\[ \text{Total collections} = 7 \times 15 \times 31 \]

### Step 4: Perform the multiplication
Now, we will calculate the total:

1. First, calculate \(7 \times 15\):
\[ 7 \times 15 = 105 \]

2. Next, multiply that result by 31:
\[ 105 \times 31 = 3255 \]

### Final Answer
Thus, the total number of different collections of books, ensuring at least one book from each subject is taken, is **3255**.

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