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Which of the following statements about combinations is always true?निम्नलिखित में से कौन-सा कथन संयोजनों के बारे में सदैव सत्य है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \(\binom{n}{r}=\binom{n}{n-r}\)\(\binom{n}{r}=\binom{n}{n-r}\)
Choosing \(r\) objects from \(n\) is equivalent to deciding which \(n-r\) objects are left out, so \(\binom{n}{r}=\binom{n}{n-r}\). Option D misses the factor \((n-r)!\). Exam tip: use this complementary-selection symmetry to identify equivalent combinations.\(\binom{n}{r}\) में चुनी गई वस्तुओं के स्थान पर छोड़ी गई \(n-r\) वस्तुओं को चुनना समान समूह निर्धारित करता है। अतः \(\binom{n}{r}=\binom{n}{n-r}\)। विकल्प D में \((n-r)!\) भी होना चाहिए। परीक्षा टिप: पूरक चयन वाली यह सममिति तुरंत पहचानें।
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