Problem 93 - Algebra and Trigonometry Real Mathematics, Real People

Question

Prove the identity. $$_{n} C_{n-1}=_{n} C_{1}$$

Step-by-Step Solution

Answer

The identity \( _nC_{n-1} = _nC_{1} \) is proven by using the combination formula. After simplification, both sides of the equation are equal, thus confirming the identity.

Step 1: Write the formula for combination

First, write the formula for combinations for each side of the identity separately: \( _nC_{n-1} = \frac{n!}{(n-1)!(n-(n-1))!} = \frac{n!}{(n-1)!1!} \) and \( _nC_{1} = \frac{n!}{1!(n-1)!} \).

Step 2: Simplify

Simplify the equations. This yields \( _nC_{n-1} = \frac{n!}{(n-1)!} \) and \( _nC_{1} = \frac{n!}{(n-1)!} \).

Step 3: Compare results

After simplification, both sides of the identity are equal. Thus, it is proven that \( _nC_{n-1} = _nC_{1} \).