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What is the simplified form of \(\frac{(n+4)!}{(n+2)!}-\frac{(n+3)!}{(n+1)!}\)?\(\frac{(n+4)!}{(n+2)!}-\frac{(n+3)!}{(n+1)!}\) का सरल रूप क्या है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \(2(n+3)\)\(2(n+3)\)
\(\frac{(n+4)!}{(n+2)!}=(n+4)(n+3)\) and \(\frac{(n+3)!}{(n+1)!}=(n+3)(n+2)\). Therefore, the expression becomes \((n+3)[(n+4)-(n+2)] = 2(n+3)\). \(2n+4\) is incorrect because \(2(n+3)=2n+6\). Exam tip: In factorial quotients, cancel the common factorial part and write the remaining consecutive factors.\(\frac{(n+4)!}{(n+2)!}=(n+4)(n+3)\) तथा \(\frac{(n+3)!}{(n+1)!}=(n+3)(n+2)\)। अतः व्यंजक \((n+3)[(n+4)-(n+2)] = 2(n+3)\) होगा। \(2n+4\) गलत है, क्योंकि सही गुणनखंड \(2(n+3)=2n+6\) है। परीक्षा टिप: फैक्टोरियल के भाग में समान फैक्टोरियल पदों को काटकर शेष क्रमागत गुणनखंड लिखें।
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