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Question
Show that
`("n"!)/("r"!("n" - "r")!) + ("n"!)/(("r" - 1)!("n" - "r" + 1)!) = (("n" + 1)!)/("r"!("n" - "r" + 1)!`
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Solution
L.H.S = `("n"!)/("r"!("n" - "r")!) + ("n"!)/(("r" - 1)!("n" - "r" + 1)!)`
`=("n"!)/("r"("r" - 1)!("n" - "r")!) + ("n"!)/(("r" - 1)! xx ("n" - "r" + 1)("n" - "r")!`
= `("n"!)/(("r" - 1)!("n" - "r")!) [1/"r" + 1/("n" - "r" + 1)]`
= `("n"!)/(("r" - 1)!("n" - "r")!) [("n" - "r" + 1 + "r")/("r"("n" - "r" + 1))]`
= `("n"!.("n" + 1))/["r"("r" - 1)!("n" - "r" + 1)("n" - "r")!)`
= `(("n" + 1)!)/("r"!("n" - "r" + 1)!]` = R.H.S.
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