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Question:

Find the coefficient of x¹² in the expansion of (1 – x⁶)⁴(1 – x)^{– 4}

Explanation:

(1 – x⁶)⁴ (1 – x)^–4

= (1 – 4x⁶ + 6x¹² – 4x¹⁸ + x²⁴)(1 – x)^–4

To find the coefficient of x¹² in the given expression, we need to find the coefficients of x¹², x⁶ and x⁰ terms in (1 – x)^–4

We use the Binomial Theorem for negative coefficients.

Coefficient of x¹²
= $\frac{(-4)(-4-1)(-4-1-2)..(-4-11)}{12!}$

= $\frac{4\times 5\times 6\times ...\times 15}{12!}$ = $\frac{13\times 14\times 15}{1\times 2\times 3}$

= 35 × 13

Coefficient of x⁶
= $\frac{(-4)(-4-1)...(-4-5)}{6!}$

= $\frac{7\times 8\times 9}{3!}$ = 84

The coefficient of x⁰ is 1.

∴ The coefficient of x¹² is 35 × 13 + (– 4) × 84 + 6

= 125

Hence, option (c).