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Chapter 11: Problem 8

Use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \({ }_{9} C_{5}\)

Short Answer

Expert verified
The value of \({ }_{9} C_{5}\) is 126.

Step by step solution

01

Identify the values of n and r

In this problem, n is given as 9 and r as 5.
02

Use the Binomial Coefficient Formula

The binomial coefficient \({ }_{n} C_{r}\) is calculated as \( \frac{n!}{r!(n-r)!} \), so we need to substitute 9 and 5 into the formula: \({ }_{9} C_{5} = \frac{9!}{5!(9-5)!}\). Now we have to solve this expression.
03

Calculate Factorials

First, we need to calculate the factorials in the expression: \(9! = 9*8*7*6*5*4*3*2*1 = 362880\), \(5! = 5*4*3*2*1 = 120\), and \((9-5)! = 4! = 4*3*2*1 = 24\)
04

Substitute Factorial Values into the Formula

Now we need to substitute the factorial values into the formula \({ }_{9} C_{5} = \frac{9!}{5!(9-5)!} = \frac{362880}{120*24}\)
05

Complete the Calculation

After substituting the factorial values into the formula and simplifying the expression, we get \({ }_{9} C_{5} = \frac{362880}{2880} = 126\)

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