determine the twelfth term of the following expression without expanding the binomial.

\(3x - y)^{16}\

Question

determine the twelfth term of the following expression without expanding the binomial.

\(3x - y)^{16}\

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Binomial Theorem", "Binomial Expansion" ], "new_concepts": [], "current_concepts": [ "Binomial Theorem", "Binomial Expansion" ] } ## Identify binomial parameters and term index $$ \begin{aligned} &(a + b)^n = (3x - y)^{16} \ &a = 3x, \quad b = -y, \quad n = 16 \ &T_{k+1} = \binom{n}{k} a^{n-k} b^k \ &k + 1 = 12 \implies k = 11 \end{aligned} $$ ## Calculate binomial coefficient and substitute values $$ \begin{aligned} \binom{16}{11} &= \frac{16!}{11! \cdot 5!} = \frac{16 \cdot 15 \cdot 14 \cdot 13 \cdot 12}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 4368 \ T_{12} &= 4368 \cdot (3x)^{16-11} \cdot (-y)^{11} \ T_{12} &= 4368 \cdot (3x)^5 \cdot (-y)^{11} \end{aligned} $$ ## Simplify the expression $$ \begin{aligned} 3^5 &= 243 \ T_{12} &= 4368 \cdot 243 \cdot x^5 \cdot (-1) \cdot y^{11} \ T_{12} &= -1,061,424x^5y^{11} \end{aligned} $$ $-1,061,424x^5y^{11}$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Binomial Theorem" ] }

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Question

Explanation:

Identify binomial parameters and term index

Calculate binomial coefficient and substitute values

Simplify the expression

Answer: