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Chapter 11: Problem 24

Evaluate each expression. \(1-\frac{{ }_{5} P_{3}}{{ }_{10} P_{4}}\)

Short Answer

Expert verified
The value of the given expression is 0.9881.

Step by step solution

01

Understanding Permutations

The permutation notation is given by \(P(n, k)\) where \(n\) is the number of items to choose from, and you need to choose \(k\) of them, so the order matters. The formula for permutation is \(P(n, k) = \frac{n!}{(n-k)!}\) where '!' denotes the factorial, i.e., \(n! = n*(n-1)*(n-2)*...*3*2*1\).
02

Calculate the Numerator

Firstly, calculate the numerator, which is \( { }_{5} P_{3} \). Using the formula \(P(n, k) = \frac{n!}{(n-k)!}\), where \(n=5\) and \(k=3\), the permutation \( { }_{5} P_{3} \) becomes \( \frac{5!}{(5-3)!} = \frac{5*4*3*2*1}{2*1} = 60.\)
03

Calculate the Denominator

Similarly, calculate the denominator which is \( { }_{10} P_{4} \). With the value of \(n=10\) and \(k=4\), it turns into \( \frac{10!}{(10-4)!} = \frac{10*9*8*7*6*5*4*3*2*1}{6*5*4*3*2*1} = 5040.\)
04

Perform Division

Now after calculating both the numerator and the denominator, we can perform the division operation as given in the problem \(\frac{60}{5040} = 0.0119.\)
05

Subtract from 1

The last step is to subtract this result from 1 as per the original equation. So, \(1-0.0119 = 0.9881.\)

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