Question

Show that: `((2"n")!)/("n"!)` = 2ⁿ(2n – 1)(2n –  3)....5.3.1

Answer 1

L.H.S. = `((2"n")!)/("n"!)`

`= ((2"n")(2"n"-1)(2"n"-2)(2"n"-3)(2"n"-4) ...6xx5xx4xx3xx 2xx1)/("n"!)`

`=((2"n")(2"n" - 1)[2("n" - 1)](2"n" - 3)[2("n" - 2)]...(2xx3)xx5xx(2xx2)xx3xx(2xx1)xx1)/("n"!)`

`=(2^"n"["n"("n"-1)("n"-2)....3.2.1][(2"n"-1)(2"n"-3)...5.3.1])/("n"!)`

= `(2^"n"("n"!)(2"n"-1)(2"n"-3)...5.3.1)/"n!"`

= 2ⁿ(2n −1)(2n −3)...5.3.1
= R.H.S.