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प्रश्न
Show that `((2"n")!)/("n"!)` = 2n (2n – 1)(2n – 3) ... 5.3.1
उत्तर
L.H.S. = `((2"n")!)/("n"!)`
= `(2"n"(2"n" - 1)(2"n" - 2)(2"n" - 3)(2"n" - 4) ... 5.4.3.2.1)/("n"!)`
= `([2"n"(2"n" - 2)(2"n" - 4) ... 6.4.2][(2"n" - 1)(2"n" - 3) ... 5.3.1])/("n"!)`
= `([2("n")*2("n" - 1)*2("n" - 2) ... 2(3)*2(2)*2(1)]*[(2"n" - 1)(2"n" - 3) ... 5.3.1])/("n"!)`
= `([2^"n"("n")("n" - 1)("n" - 2) ... 3.2.1][(2"n" - 1)(2"n" - 3) ... 5.3.1])/("n"!)`
= `(2^"n" xx "n"![(2"n" - 1)(2"n" - 3) ... 5.3.1])/("n"!)`
= 2n(2n – 1)(2n – 3) ... 5.3.1
= R.H.S.
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