determine the sixth term of the following expression without expanding the binomial.
\(x + y)^{10}\

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Binomial Theorem", "Binomial Expansion", "Binomial Coefficient" ], "new_concepts": [], "current_concepts": [ "Binomial Theorem", "Binomial Expansion", "Binomial Coefficient" ] } ## Identify the binomial parameters and the term formula $$ \begin{aligned} &(a + b)^n = (x + y)^{10} \implies a = x,\ b = y,\ n = 10\ &T_{k+1} = \binom{n}{k} a^{n-k} b^k \end{aligned} $$ ## Determine the index for the sixth term $$ \begin{aligned} &k + 1 = 6 \implies k = 5\ &T_6 = \binom{10}{5} x^{10-5} y^5 \end{aligned} $$ ## Calculate the binomial coefficient and simplify $$ \begin{aligned} &\binom{10}{5} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252\ &T_6 = 252x^5y^5 \end{aligned} $$ $252x^5y^5$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Binomial Theorem" ] }

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Explanation:

Identify the binomial parameters and the term formula

Determine the index for the sixth term

Calculate the binomial coefficient and simplify

Answer: