The number of positive integers satisfying the inequality
`""^(n+1)C_(3) - ""^(n-1) C_(2) le 100` is

Options

A. nine
B. eight
C. five
D. None of these

Answer

To solve the inequality \( \binom{n+1}{3} - \binom{n-1}{2} \leq 100 \), we will follow these steps:

### Step 1: Write down the inequality
The given inequality is:
\[ \binom{n+1}{3} - \binom{n-1}{2} \leq 100 \]

### Step 2: Expand the binomial coefficients
Using the formula for binomial coefficients \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), we can expand the terms:
\[ \binom{n+1}{3} = \frac{(n+1)n(n-1)}{3!} = \frac{(n+1)n(n-1)}{6} \]
\[ \binom{n-1}{2} = \frac{(n-1)(n-2)}{2!} = \frac{(n-1)(n-2)}{2} \]

### Step 3: Substitute the expanded forms into the inequality
Substituting these into the inequality gives:
\[ \frac{(n+1)n(n-1)}{6} - \frac{(n-1)(n-2)}{2} \leq 100 \]

### Step 4: Simplify the inequality
To simplify, we can find a common denominator, which is 6:
\[ \frac{(n+1)n(n-1)}{6} - \frac{3(n-1)(n-2)}{6} \leq 100 \]
Combining the fractions:
\[ \frac{(n+1)n(n-1) - 3(n-1)(n-2)}{6} \leq 100 \]
Multiplying through by 6 (since 6 is positive, the inequality direction remains the same):
\[ (n+1)n(n-1) - 3(n-1)(n-2) \leq 600 \]

### Step 5: Factor out common terms
Factoring out \( (n-1) \):
\[ (n-1) \left( (n+1)n - 3(n-2) \right) \leq 600 \]
Expanding the expression inside the parentheses:
\[ (n-1) \left( n^2 + n - 3n + 6 \right) \leq 600 \]
This simplifies to:
\[ (n-1)(n^2 - 2n + 6) \leq 600 \]

### Step 6: Solve for \( n \)
Now we will test positive integer values for \( n \) to find how many satisfy this inequality.

1. **For \( n = 1 \)**:
\[ (1-1)(1^2 - 2 \cdot 1 + 6) = 0 \leq 600 \quad \text{(satisfies)} \]

2. **For \( n = 2 \)**:
\[ (2-1)(2^2 - 2 \cdot 2 + 6) = 1 \cdot 2 = 2 \leq 600 \quad \text{(satisfies)} \]

3. **For \( n = 3 \)**:
\[ (3-1)(3^2 - 2 \cdot 3 + 6) = 2 \cdot 3 = 6 \leq 600 \quad \text{(satisfies)} \]

4. **For \( n = 4 \)**:
\[ (4-1)(4^2 - 2 \cdot 4 + 6) = 3 \cdot 10 = 30 \leq 600 \quad \text{(satisfies)} \]

5. **For \( n = 5 \)**:
\[ (5-1)(5^2 - 2 \cdot 5 + 6) = 4 \cdot 16 = 64 \leq 600 \quad \text{(satisfies)} \]

6. **For \( n = 6 \)**:
\[ (6-1)(6^2 - 2 \cdot 6 + 6) = 5 \cdot 24 = 120 \leq 600 \quad \text{(satisfies)} \]

7. **For \( n = 7 \)**:
\[ (7-1)(7^2 - 2 \cdot 7 + 6) = 6 \cdot 36 = 216 \leq 600 \quad \text{(satisfies)} \]

8. **For \( n = 8 \)**:
\[ (8-1)(8^2 - 2 \cdot 8 + 6) = 7 \cdot 50 = 350 \leq 600 \quad \text{(satisfies)} \]

9. **For \( n = 9 \)**:
\[ (9-1)(9^2 - 2 \cdot 9 + 6) = 8 \cdot 66 = 528 \leq 600 \quad \text{(satisfies)} \]

10. **For \( n = 10 \)**:
\[ (10-1)(10^2 - 2 \cdot 10 + 6) = 9 \cdot 84 = 756 \not\leq 600 \quad \text{(does not satisfy)} \]

### Conclusion
The positive integers \( n \) that satisfy the inequality are \( 1, 2, 3, 4, 5, 6, 7, 8, 9 \), giving us a total of **9 positive integers**.

### Final Answer
The number of positive integers satisfying the inequality is **9**.

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