Solve the inequality `.^(x-1)C_(4)-.^(x-1)C_(3)-(5)/(2).^(x-2)C_(2)lt0,x inN`.

Explanation

To solve the inequality \( \binom{x-1}{4} - \binom{x-1}{3} - \frac{5}{2} \binom{x-2}{2} < 0 \) for \( x \in \mathbb{N} \), we will follow these steps:

### Step 1: Rewrite the Binomial Coefficients
We start by rewriting the binomial coefficients using their definitions:
- \( \binom{x-1}{4} = \frac{(x-1)(x-2)(x-3)(x-4)}{4!} = \frac{(x-1)(x-2)(x-3)(x-4)}{24} \)
- \( \binom{x-1}{3} = \frac{(x-1)(x-2)(x-3)}{3!} = \frac{(x-1)(x-2)(x-3)}{6} \)
- \( \binom{x-2}{2} = \frac{(x-2)(x-3)}{2!} = \frac{(x-2)(x-3)}{2} \)

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