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

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

$\frac{^{n}{}{C}_r}{^{n - 1}{}{C}_{r - 1}} = \frac{n}{r}$

#### Solution

$\frac{n_{C_r}}{n - 1_{C_{r - 1}}} = \frac{n}{r}$

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

∴  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 20.3 | Page 9