Tamil Nadu Board of Secondary EducationHSC Arts Class 11

Prove that if 1 ≤ r ≤ n then nnCrnrCrn×(n-1)Cr-1=(n-r+1)Cr-1 - Mathematics

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Sum

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

Concept: Combinations
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Chapter 4: Combinatorics and Mathematical Induction - Exercise 4.3 [Page 186]

APPEARS IN

Tamil Nadu Board Samacheer Kalvi Class 11th Mathematics Volume 1 and 2 Answers Guide
Chapter 4 Combinatorics and Mathematical Induction
Exercise 4.3 | Q 8 | Page 186

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