The number of positive integers satisfying the inequality `C(n+1,n-2) - C(n+1,n-1)<=100` is

Options

A. 9
B. 8
C. 5
D. None of these

Explanation

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

### Step 1: Write the combinations in terms of factorials
The combination \( C(n, r) \) is defined as:
\[ C(n, r) = \frac{n!}{r!(n-r)!} \]
Thus, we can express \( C(n+1, n-2) \) and \( C(n+1, n-1) \) as follows:

\[ C(n+1, n-2) = \frac{(n+1)!}{(n-2)!(3!)} = \frac{(n+1)n(n-1)}{6} \]
\[ C(n+1, n-1) = \frac{(n+1)!}{(n-1)!(2!)} = \frac{(n+1)n}{2} \]

### Step 2: Substitute the combinations into the inequality
Now, substituting these values into the inequality:
\[ \frac{(n+1)n(n-1)}{6} - \frac{(n+1)n}{2} \leq 100 \]

### Step 3: Simplify the inequality
To simplify the left-hand side, we can find a common denominator, which is 6:
\[ \frac{(n+1)n(n-1)}{6} - \frac{3(n+1)n}{6} \leq 100 \]
This simplifies to:
\[ \frac{(n+1)n(n-1) - 3(n+1)n}{6} \leq 100 \]
\[ \frac{(n+1)n((n-1) - 3)}{6} \leq 100 \]
\[ \frac{(n+1)n(n-4)}{6} \leq 100 \]

### Step 4: Multiply both sides by 6
Multiplying both sides by 6 (since 6 is positive):
\[ (n+1)n(n-4) \leq 600 \]

### Step 5: Rearranging the inequality
This can be rewritten as:
\[ n(n^2 - 3n - 4) \leq 600 \]
or
\[ n^3 - 3n^2 - 4n - 600 \leq 0 \]

### Step 6: Find the roots of the cubic equation
To find the values of \( n \) satisfying the inequality, we will check integer values starting from \( n = 2 \) (since \( n \) must be a positive integer and \( n-2 \geq 0 \)).

1. For \( n = 2 \):
\[ 2^3 - 3(2^2) - 4(2) - 600 = 8 - 12 - 8 - 600 = -612 \quad (\text{valid}) \]

2. For \( n = 3 \):
\[ 3^3 - 3(3^2) - 4(3) - 600 = 27 - 27 - 12 - 600 = -612 \quad (\text{valid}) \]

3. For \( n = 4 \):
\[ 4^3 - 3(4^2) - 4(4) - 600 = 64 - 48 - 16 - 600 = -600 \quad (\text{valid}) \]

4. For \( n = 5 \):
\[ 5^3 - 3(5^2) - 4(5) - 600 = 125 - 75 - 20 - 600 = -570 \quad (\text{valid}) \]

5. For \( n = 6 \):
\[ 6^3 - 3(6^2) - 4(6) - 600 = 216 - 108 - 24 - 600 = -516 \quad (\text{valid}) \]

6. For \( n = 7 \):
\[ 7^3 - 3(7^2) - 4(7) - 600 = 343 - 147 - 28 - 600 = -432 \quad (\text{valid}) \]

7. For \( n = 8 \):
\[ 8^3 - 3(8^2) - 4(8) - 600 = 512 - 192 - 32 - 600 = -312 \quad (\text{valid}) \]

8. For \( n = 9 \):
\[ 9^3 - 3(9^2) - 4(9) - 600 = 729 - 243 - 36 - 600 = -150 \quad (\text{valid}) \]

9. For \( n = 10 \):
\[ 10^3 - 3(10^2) - 4(10) - 600 = 1000 - 300 - 40 - 600 = 60 \quad (\text{not valid}) \]

### Conclusion
The valid values of \( n \) are \( 2, 3, 4, 5, 6, 7, 8, 9 \), which gives us a total of **8 positive integers** satisfying the inequality.