Given the meaning and value of the symbol in the following
` ""^(7) P_3`

Answer

To solve the problem, we need to understand the notation \( nPr \) and then calculate the value of \( 7P3 \).

### Step-by-Step Solution:

1. **Understanding the Notation**:
- The notation \( nPr \) represents the number of permutations of \( n \) objects taken \( r \) at a time.
- Here, \( n \) is the total number of objects, and \( r \) is the number of objects to be arranged.

2. **Identifying Values**:
- In the expression \( 7P3 \):
- \( n = 7 \) (total objects)
- \( r = 3 \) (objects taken at a time)

3. **Formula for Permutations**:
- The formula for permutations is given by:
\[ nPr = \frac{n!}{(n - r)!} \]
- Where \( n! \) (n factorial) is the product of all positive integers up to \( n \).

4. **Applying the Formula**:
- Substitute \( n = 7 \) and \( r = 3 \) into the formula:
\[ 7P3 = \frac{7!}{(7 - 3)!} = \frac{7!}{4!} \]

5. **Calculating Factorials**:
- Calculate \( 7! \) and \( 4! \):
- \( 7! = 7 \times 6 \times 5 \times 4! \)
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

6. **Simplifying the Expression**:
- Substitute \( 7! \) into the expression:
\[ 7P3 = \frac{7 \times 6 \times 5 \times 4!}{4!} \]
- The \( 4! \) in the numerator and denominator cancels out:
\[ 7P3 = 7 \times 6 \times 5 \]

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

8. **Conclusion**:
- Therefore, the value of \( 7P3 \) is \( 210 \).

### Final Answer:
- The meaning of \( 7P3 \) is the number of permutations of 7 objects taken 3 at a time, and its value is \( 210 \).