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