There are 7 candidates for 3 posts. In how many ways can the posts be filled ?

Options

A. 120
B. 130
C. 100
D. 210

Answer

To solve the problem of how many ways 3 posts can be filled by 7 candidates, we can use the concept of permutations since the order in which the candidates are selected for the posts matters.

### Step-by-Step Solution:

1. **Identify the total number of candidates (n) and the number of posts (r)**:
- Total candidates (n) = 7
- Total posts (r) = 3

2. **Use the permutation formula**:
- The formula for permutations is given by:
\[ P(n, r) = \frac{n!}{(n - r)!} \]
- Here, \( n! \) represents the factorial of n, which is the product of all positive integers up to n.

3. **Substitute the values into the formula**:
- Substitute n = 7 and r = 3 into the permutation formula:
\[ P(7, 3) = \frac{7!}{(7 - 3)!} = \frac{7!}{4!} \]

4. **Calculate the factorials**:
- Calculate \( 7! \) (which is \( 7 \times 6 \times 5 \times 4! \)):
\[ 7! = 7 \times 6 \times 5 \times 4! \]
- Since \( 4! \) appears in both the numerator and the denominator, we can cancel it out:
\[ P(7, 3) = \frac{7 \times 6 \times 5 \times 4!}{4!} = 7 \times 6 \times 5 \]

5. **Perform the multiplication**:
- Now, calculate \( 7 \times 6 \times 5 \):
\[ 7 \times 6 = 42 \]
\[ 42 \times 5 = 210 \]

6. **Conclusion**:
- Therefore, the total number of ways to fill the 3 posts from 7 candidates is **210**.

### Final Answer:
The number of ways to fill the posts is **210**.