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Chapter 9: Problem 51

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

Short Answer

Expert verified
The answer is 10

Step by step solution

01

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
02

Substitute the values into the formula

Substitute n=5 and r=2 into the combination formula. The formula becomes: \(_{5} C_{2} = \frac{5!}{2!(5-2)!}\)
03

Simplify the Factorials at top and bottom

5! = 5 脳 4 脳 3 脳 2 脳 1, 2! = 2 脳 1 and (5-2)! = 3!, which equals to 3 脳 2 脳 1. Thus, \(_{5} C_{2} = \frac{5 脳 4 脳 3 脳 2 脳 1}{2 脳 1 脳 3 脳 2 脳 1}\)
04

Cancel out repeat values

After canceling common terms from the top and bottom, the value becomes \(_{5} C_{2} = 5 脳 4 / 2 脳 1\)
05

Final Step: Calculate the value

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

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