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Prove that if 1 ≤ r ≤ n then `"n" xx ""^(("n" - 1))"C"_("r" - 1) = ""^(("n" - "r" + 1))"C"_("r" - 1)`

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#### Solution

To Prove `"n"[""^("n" - 1)"C"_("r" - 1)] = ""^(("n" - "r" + 1))[""^"n""C"_("r" - 1)]`

L.H.S = `"n"[(("n" - 1)!)/(("r" - 1)!("n" - 1 - ("r" - 1))!("n" - 1 - "r" + 1))]`

= `(""("n" - 1)!)/(("r" - 1)!("n" - "r")!) = ("n"!)/(("r" - 1)!("n" - "r")!)` .....(1)

R.H.S = `""^(("n" - "r" + 1))[""^"n""C"_("r" - 1)]`

= `("n" - "r" + 1)[("n"!)/(("r" - 1)!("n" - "r" - 1)!("n" - "r"+ 1))]`

= `("n" - "r" + 1)[("n"!)/(("r" - 1)!("n" -"r" + 1)!)]`

= `(("n" - "r" + 1)"n"!)/(("r" - 1)!("n" - "r" + 1)("n" - "r")!)`

= `("n"!)/(("r" - 1)!("n" - "r")!)` ......(2)

(1) = (2)

⇒ L.H.S = R.H.S

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