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Let R and N Be Positive Integers Such that 1 ≤ R ≤ N. Then Prove the Following: N · N − 1cr − 1 = (N − R + 1) Ncr − 1 - Mathematics

Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:
n · n − 1Cr − 1 = (n − r + 1) nCr − 1

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Solution

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

∴ LHS = RHS

Concept: Factorial N (N!) Permutations and Combinations
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 17 Combinations
Exercise 17.1 | Q 20.2 | Page 9
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