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Let R and N Be Positive Integers Such that 1 ≤ R ≤ N. Then Prove the Following: N C R N C R − 1 = N − R + 1 R - Mathematics

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Question

Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:

\[\frac{^{n}{}{C}_r}{^{n}{}{C}_{r - 1}} = \frac{n - r + 1}{r}\]
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Solution

\[\frac{{}^n C_r}{{}^n C_{r - 1}} = \frac{n - r + 1}{r}\]

\[LHS = \frac{{}^n C_r}{{}^n C_{r - 1}} \]
\[ = \frac{n!}{r! \left( n - r \right)!} \times \frac{\left( r - 1 \right)! \left( n - r + 1 \right)!}{n!} \]
\[ = \frac{\left( n - r + 1 \right) \left( n - r \right)! \left( r - 1 \right)!}{r \left( r - 1 \right)! \left( n - r \right)!}\]
\[ = \frac{n - r + 1}{r} = RHS\]

∴\[LHS = RHS\]

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Factorial N (N!) Permutations and Combinations
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Chapter 17: Combinations - Exercise 17.1 [Page 9]

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RD Sharma Mathematics [English] Class 11
Chapter 17 Combinations
Exercise 17.1 | Q 20.1 | Page 9

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