Verify that `2 ""^(7)C_(4)=""^(8)C_(4)`.

Answer

To verify that \( 2 \times \binom{7}{4} = \binom{8}{4} \), we will compute both sides using the formula for combinations.

### Step-by-Step Solution:

1. **Understanding the Combination Formula**:
The formula for combinations is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

2. **Calculate \( \binom{7}{4} \)**:
Using the formula, we can calculate \( \binom{7}{4} \):
\[ \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7!}{4! \cdot 3!} \]

3. **Expanding the Factorials**:
We can expand \( 7! \) as follows:
\[ 7! = 7 \times 6 \times 5 \times 4! \]
Thus, we have:
\[ \binom{7}{4} = \frac{7 \times 6 \times 5 \times 4!}{4! \cdot 3!} \]

4. **Canceling \( 4! \)**:
The \( 4! \) in the numerator and denominator cancels out:
\[ \binom{7}{4} = \frac{7 \times 6 \times 5}{3!} \]

5. **Calculating \( 3! \)**:
We know that \( 3! = 6 \), so:
\[ \binom{7}{4} = \frac{7 \times 6 \times 5}{6} = 7 \times 5 = 35 \]

6. **Calculating \( 2 \times \binom{7}{4} \)**:
Now, we calculate:
\[ 2 \times \binom{7}{4} = 2 \times 35 = 70 \]

7. **Calculate \( \binom{8}{4} \)**:
Now, we compute \( \binom{8}{4} \):
\[ \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4! \cdot 4!} \]

8. **Expanding \( 8! \)**:
We can expand \( 8! \) as follows:
\[ 8! = 8 \times 7 \times 6 \times 5 \times 4! \]
Thus, we have:
\[ \binom{8}{4} = \frac{8 \times 7 \times 6 \times 5 \times 4!}{4! \cdot 4!} \]

9. **Canceling \( 4! \)**:
The \( 4! \) in the numerator and denominator cancels out:
\[ \binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4!} \]

10. **Calculating \( 4! \)**:
Again, \( 4! = 24 \), so:
\[ \binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{24} \]

11. **Calculating the Numerator**:
First, calculate \( 8 \times 7 \times 6 \times 5 \):
\[ 8 \times 7 = 56 \]
\[ 56 \times 6 = 336 \]
\[ 336 \times 5 = 1680 \]

12. **Final Calculation**:
Now divide by \( 24 \):
\[ \binom{8}{4} = \frac{1680}{24} = 70 \]

13. **Conclusion**:
Since both calculations yield the same result:
\[ 2 \times \binom{7}{4} = 70 \quad \text{and} \quad \binom{8}{4} = 70 \]
We have verified that \( 2 \times \binom{7}{4} = \binom{8}{4} \).