There are 3 books of mathematics, 4 of science and 5 of literature. How many different collections can be made such that each collection consists of, atleast one book of literature.

Options

A. a. 6368
B. b. 3968
C. c. 3600
D. d. 4400

Answer

To solve the problem of how many different collections can be made such that each collection consists of at least one book of literature, we can follow these steps:

### Step 1: Determine the number of ways to select books from each category
We have:
- 3 Mathematics books
- 4 Science books
- 5 Literature books

For each book, we have two choices: either to select the book or not to select it.

### Step 2: Calculate the total selections for each category
- For Mathematics books: The number of ways to select from 3 books is \(2^3\).
- For Science books: The number of ways to select from 4 books is \(2^4\).
- For Literature books: The number of ways to select from 5 books is \(2^5\).

Calculating these:
- \(2^3 = 8\) (for Mathematics)
- \(2^4 = 16\) (for Science)
- \(2^5 = 32\) (for Literature)

### Step 3: Calculate the total combinations without restrictions
The total number of combinations of selecting books from all categories (including the case where no literature books are selected) is:
\[ Total = 2^3 \times 2^4 \times 2^5 = 8 \times 16 \times 32 \]

### Step 4: Calculate the total combinations
Calculating the product:
\[ Total = 8 \times 16 = 128 \]
\[ Total = 128 \times 32 = 4096 \]

### Step 5: Exclude the case where no literature books are selected
Now, we need to exclude the scenario where no literature books are selected. If no literature books are selected, we can only select from Mathematics and Science books:
- The number of ways to select from Mathematics and Science (without selecting any Literature books) is:
\[ 2^3 \times 2^4 = 8 \times 16 = 128 \]

### Step 6: Calculate the valid combinations
Now, we subtract the case where no Literature books are selected from the total combinations:
\[ Valid\ Combinations = Total - (Combinations\ without\ Literature) = 4096 - 128 = 3968 \]

### Final Answer
Thus, the total number of different collections that can be made such that each collection consists of at least one book of literature is **3968**.

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