Binomial theorem |Hard question of binomial theorem |Easy approach |shortcut|26 January 2026

Описание: Easy way to find the coefficient of binomial expansion .
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The video explains how to find the coefficient of a specific term in a complex algebraic expression involving the binomial theorem (0:00-0:29). The target term is x raised to the power of n² + n - 14 / 2 (0:06).

Here's a breakdown of the solution approach:
• Understanding the target power (0:34-1:32): The expression's power can be simplified using the sum of the first 'n' natural numbers (σ n) and then subtracting 7. This means the goal is to find the coefficient when the x term with a power of 7 is not included in the product (1:45-1:48).
• Identifying single terms to exclude (2:35-3:33): The primary step is to find the coefficient when the x⁷ term itself is excluded. This directly results in a coefficient of -7.
• Excluding pairs of terms (3:49-6:11): The next step involves identifying pairs of x terms whose exponents sum to 7. By excluding these x terms and instead taking their constant counterparts, additional coefficients are found:
• Excluding x¹ and x⁶ leads to a coefficient of +6 (4:50).
• Excluding x² and x⁵ leads to a coefficient of +10 (5:39-5:41).
• Excluding x³ and x⁴ leads to a coefficient of +12 (6:06-6:11).
• Excluding triplets of terms (6:27-7:17): Finally, the video explores triplets of x terms whose exponents sum to 7. By taking their constant parts, another coefficient is determined:
• Excluding x¹, x², and x⁴ results in a coefficient of -8 (7:17).
• Final Calculation (7:21-7:35): Summing all the obtained coefficients (-7, +6, +10, +12, -8) gives the final answer, which is 13.