Problem 51 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate \(_{n} C_{r}\) using the formula from this section. $$_{5} C_{2}$$

Step-by-Step Solution

Answer

The answer is 10

Step 1: Begin by identifying the formula

The combination formula is represented by \(_{n} C_{r} = \frac{n!}{r!(n-r)!}\). For the problem at hand, n=5 and r=2

Step 2: Substitute the values into the formula

Substitute n=5 and r=2 into the combination formula. The formula becomes: \(_{5} C_{2} = \frac{5!}{2!(5-2)!}\)

Step 3: Simplify the Factorials at top and bottom

5! = 5 × 4 × 3 × 2 × 1, 2! = 2 × 1 and (5-2)! = 3!, which equals to 3 × 2 × 1. Thus, \(_{5} C_{2} = \frac{5 × 4 × 3 × 2 × 1}{2 × 1 × 3 × 2 × 1}\)

Step 4: Cancel out repeat values

After canceling common terms from the top and bottom, the value becomes \(_{5} C_{2} = 5 × 4 / 2 × 1\)

Step 5: Final Step: Calculate the value

Resolve the top and bottom part respectively to get: \(_{5} C_{2} = \frac{20}{2} = 10\)