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If (\binom{n}{5}=\binom{n}{9}), what will be the value of (n)?यदि (\binom{n}{5}=\binom{n}{9}) है तो (n) का मान क्या होगा?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: 1414
Use the symmetry property of combinations: \(\binom{n}{r}=\binom{n}{n-r}\). For \(\binom{n}{5}=\binom{n}{9}\), 9 must be the complement of 5. Thus, \(n-5=9\), so \(n=14\). For option 13, \(n-5=8\), so it does not satisfy the equality. Exam tip: when two combinations are equal, first check whether their lower indices are complementary.संयोजन का सममिति गुण है: \(\binom{n}{r}=\binom{n}{n-r}\)। यहाँ \(\binom{n}{5}=\binom{n}{9}\) होने के लिए 9, 5 का पूरक होना चाहिए। अतः \(n-5=9\), इसलिए \(n=14\)। विकल्प 13 में \(n-5=8\) होगा, इसलिए वह सही नहीं है। परीक्षा टिप: समान संयोजनों में पहले \(r\) और \(n-r\) की सममिति जाँचें।
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