In how many ways can 4 books be selected out of 16 books on different subjects?

Options

A. 1208
B. 1820
C. 1296
D. 1860

Answer

To solve the problem of selecting 4 books from a total of 16 books, we will use the concept of combinations. The formula for combinations is given by:

\[ nCk = \frac{n!}{k!(n-k)!} \]

where:
- \( n \) is the total number of items (in this case, 16 books),
- \( k \) is the number of items to choose (in this case, 4 books),
- \( ! \) denotes factorial, which is the product of all positive integers up to that number.

### Step-by-Step Solution:

1. **Identify the values of n and k**:
- Here, \( n = 16 \) (the total number of books) and \( k = 4 \) (the number of books to select).

2. **Apply the combination formula**:
\[ 16C4 = \frac{16!}{4!(16-4)!} = \frac{16!}{4! \cdot 12!} \]

3. **Simplify the factorials**:
- We can expand \( 16! \) as follows:
\[ 16! = 16 \times 15 \times 14 \times 13 \times 12! \]
- Therefore, substituting this into the combination formula gives:
\[ 16C4 = \frac{16 \times 15 \times 14 \times 13 \times 12!}{4! \cdot 12!} \]
- The \( 12! \) cancels out:
\[ 16C4 = \frac{16 \times 15 \times 14 \times 13}{4!} \]

4. **Calculate \( 4! \)**:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

5. **Substitute \( 4! \) into the equation**:
\[ 16C4 = \frac{16 \times 15 \times 14 \times 13}{24} \]

6. **Calculate the numerator**:
- First, calculate \( 16 \times 15 \times 14 \times 13 \):
\[ 16 \times 15 = 240 \]
\[ 240 \times 14 = 3360 \]
\[ 3360 \times 13 = 43680 \]

7. **Divide by \( 4! \)**:
\[ 16C4 = \frac{43680}{24} \]
- Performing the division:
\[ 43680 \div 24 = 1820 \]

8. **Final answer**:
- Therefore, the number of ways to select 4 books from 16 is \( 1820 \).

### Conclusion:
The total number of ways to select 4 books from 16 books is **1820**.