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Chapter 11: Problem 22

Evaluate each expression. \(\frac{{ }_{20} P_{2}}{2 !}-{ }_{20} C_{2}\)

Short Answer

Expert verified
The evaluated difference between the permutation and the combination is 0.

Step by step solution

01

Evaluate the Permutation

First, we will evaluate the permutation, \( _{20}P_{2} \). A permutation of 'n' objects taken 'r' at a time is given by the equation \( _nP_r = \frac{n!}{(n-r)!} \). So the permutation \( _{20}P_{2} \) is evaluated as \( _{20}P_{2} = \frac{20!}{(20-2)!} = \frac{20!}{18!} = 380 \)
02

Evaluate the Combination

Now, we will evaluate the combination, \( _{20}C_{2} \). A combination of 'n' objects taken 'r' at a time is given by the equation \( _nC_r = \frac{n!}{r!(n-r)!} \). So the combination \( _{20}C_{2} \) is evaluated as \( _{20}C_{2} = \frac{20!}{2!(20-2)!} = \frac{20!}{2!18!} = 190 \)
03

Subtract the Combination from the Permutation

The exercise asks us to subtract the calculated combination from the calculated permutation. So now, we do \( _{20}P_{2}/2! - _{20}C_{2} \) = \( 380/2 - 190 \) = 190 - 190 = 0

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