Problem 94 - Algebra and Trigonometry Real Mathematics, Real People

Question

Prove the identity. $$_{n} C_{r}=\frac{_{n} P_{r}}{r !}$$

Step-by-Step Solution

Answer

Based on the definitions for combination and permutation, it can be proved that \( _{n} C_{r}=\frac{_{n} P_{r}}{r !} \) . The identity is proven by simplifying both sides of the equation to \( \frac{n !}{r ! (n - r) !} \).

Step 1: Understand the given identity

The problem is asking to prove the formula \( _{n} C_{r}=\frac{_{n} P_{r}}{r !} \) . Here \( _{n} C_{r} \) is a combination, which means out of n items, we are choosing r items. \( _{n} P_{r} \) is a permutation, which means out of n items, we are arranging r items. r ! (r factorial) refers to the product of all positive integers up to r.

Step 2: Use the formula for combination and permutation

The formula for combination is \( _{n} C_{r} = \frac{n !}{r !(n-r)!} \) . The formula for permutation is \( _{n} P_{r}= \frac{n !}{(n-r) !} \) . Replace these in the given formula.

Step 3: Prove the identity

Replacing the formulas in the given identity, we get \(\frac{n !}{r !(n-r)!} = \frac{\frac{n !}{(n-r)!}}{r !} \) . If we further simplify this, both sides of the equation simplify to \( \frac{n !}{r ! (n - r) !} \). Thus, proving the identity.