/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 27 Evaluate each expression. \(\f... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 27

Evaluate each expression. \(\frac{{ }_{4} C_{2} \cdot{ }_{6} C_{1}}{{ }_{18} C_{3}}\)

Short Answer

Expert verified
The evaluated expression is approximately \(0.0000379\).

Step by step solution

01

Translate the Combinations

Translate the combinations in the expression into their respective combination formula. Here is the structure \( _{n}C_{r} = \frac{n!}{r!(n-r)!}\). Thus, \( _{4}C_{2} = \frac{4!}{2!(4-2)!}\), \( _{6}C_{1} = \frac{6!}{1!(6-1)!}\), and \( _{18}C_{3} = \frac{18!}{3!(18-3)!}\). Substitute these three expressions.
02

Simplify the Factorials

Next, we simplify the factorials in the expressions from step 1. \(4! = 4*3*2*1 = 24\), \(2! = 2*1 = 2\), \(6! = 720\), \(1! = 1\), \(18! = 6402373705728000\), \(3! = 6\). After simplification, \( _{4}C_{2} = \frac{24}{2*2} = 3\), \( _{6}C_{1} = \frac{720}{1*120} = 6\), and \( _{18}C_{3} = \frac{6402373705728000}{6*2250} = 475020. Substitute the simplified values into the original expression.
03

Evaluate the Expression

Finally, substitute the values of each expression obtained above into the original combination expression. The expression becomes \( \frac{3*6}{475020}\), which evaluates to approximately \(0.0000379\).

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