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Chapter 11: Problem 15

Use the formula for \({ }_{n} C_{r}\) to evaluate each expression. \({ }_{30} C_{3}\)

Short Answer

Expert verified
The result for \({ }_{30} C_{3}\) is 4060.

Step by step solution

01

Compute the factorials

First, compute the factorial of 30, 3, and 27 as they'll be needed for the formula. Please remember that the factorial of a positive integer n, denoted by \(n!\) , is the product of all positive integers from 1 to n. Given this, we understand that \(30! = 30 * 29 * 28 * ... * 3 * 2 * 1\), \(3! = 3 * 2 * 1\), and \(27! = 27 * 26 * 25 * ... * 3 * 2 * 1\).
02

Substitute values into the formula

Next, substitute your computed values into the formula which becomes: \({ }_{30} C_{3} = \frac{30!}{3!(30-3)!} = \frac{30!}{3!*27!}\)
03

Simplify the expression

To simplify this expression, reduce the larger factorial, realize that \(30! = 30 * 29 * 28 * ... * 3 * 2 * 1\) and \(27! = 27 * 26 * 25 * ... * 3 * 2 * 1\). The common terms of \(27! * 26 * 25 * ... * 3 * 2 * 1\) in both numerator and denominator will cancel out. Hence, \({ }_{30} C_{3} = \frac{30 * 29 * 28 * ... * 3 * 2 * 1}{3 * 2 * 1 * 27 * 26 * 25 * ... * 3 * 2* 1}\) simplifies to \({ }_{30} C_{3} = \frac{30 * 29 * 28}{3 * 2 * 1}\).
04

Evaluate the result

Finally, calculate the result \({ }_{30} C_{3} = \frac{30 * 29 * 28}{3 * 2 * 1} = 4060\)

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