Problem 52 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate \(_{n} C_{r}\) using the formula from this section. $$_{6} C_{3}$$

Step-by-Step Solution

Answer

The value of \(_{6}C_{3} = 20\).

Step 1: Identify values in the combination formula

In this case, the total number of items, \( n = 6 \), and the chosen number of items, \( r = 3 \).

Step 2: Calculate the factorials

First calculate \( 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720 \), then calculate \( 3! = 3 * 2 * 1 = 6 \), and finally calculate \( (6-3)! = 3! = 6 \).

Step 3: Insert the values into the formula

Substitute the calculated values into the combination formula: \( _{6}C_{3} = \frac{6!}{3!(6-3)!} = \frac{720}{6*6} \).

Step 4: Simplify the equation

Solve the equation to get the final result: \( _{6}C_{3} = \frac{720}{36} = 20 \).