In a shelf there are 6 physics, 4 chemistry and 3 mathematics books. How many combinations are there if (i) books of same subject are different ? (ii) books of same subject are identical?

Answer

To solve the problem step by step, we will address both parts of the question separately.

### Part (i): Books of the same subject are different

1. **Identify the total number of books**:
- Physics books = 6
- Chemistry books = 4
- Mathematics books = 3
- Total books = 6 + 4 + 3 = 13

2. **Calculate the total combinations**:
- Each book can either be selected or not selected. Therefore, for each of the 13 books, there are 2 choices (select or not select).
- The total number of ways to select any combination of books (including selecting none) is \(2^{13}\).

3. **Exclude the case where no books are selected**:
- Since we want at least one book to be selected, we subtract the case where no books are selected (which is 1 way).
- Therefore, the total combinations where at least one book is selected is \(2^{13} - 1\).

4. **Calculate \(2^{13}\)**:
- \(2^{13} = 8192\).

5. **Final calculation**:
- Total combinations = \(8192 - 1 = 8191\).

### Answer for Part (i):
The total number of combinations if books of the same subject are different is **8191**.

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### Part (ii): Books of the same subject are identical

1. **Identify the number of books per subject**:
- Physics books = 6 (identical)
- Chemistry books = 4 (identical)
- Mathematics books = 3 (identical)

2. **Use the formula for combinations of identical items**:
- The formula for selecting at least one book from each subject is given by \((p + 1)(q + 1)(r + 1) - 1\), where \(p\), \(q\), and \(r\) are the number of identical books in each subject.
- Here, \(p = 6\) (Physics), \(q = 4\) (Chemistry), \(r = 3\) (Mathematics).

3. **Calculate the combinations**:
- \((6 + 1)(4 + 1)(3 + 1) - 1\)
- This simplifies to \(7 \times 5 \times 4 - 1\).

4. **Perform the multiplication**:
- \(7 \times 5 = 35\)
- \(35 \times 4 = 140\).

5. **Final calculation**:
- Total combinations = \(140 - 1 = 139\).

### Answer for Part (ii):
The total number of combinations if books of the same subject are identical is **139**.

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