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

Sum

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

#### 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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