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Chapter 11: Problem 21

In Exercises 21-28, evaluate each expression. \(\frac{{ }_{7} P_{3}}{3 !}-{ }_{7} C_{3}\)

Short Answer

Expert verified
The solution for the given expression \(\frac{_{7}P_{3}}{3 !}-{ }_{7}C_{3}\) is 315.

Step by step solution

01

Calculate Permutation

First, compute the permutation \(_{7}P_{3}\) using the formula \(\frac{n!}{(n-r)!}\). Here, n=7 and r=3. Thus, \(_{7}P_{3} = \frac{7!}{(7-3)!}\). After calculation, we find that \(_{7}P_{3} = 2100\)
02

Calculate Factorial

Next, calculate the factorial for 3, denoted as 3!. So, 3! = 3 * 2 * 1 = 6.
03

Division of Permutation and Factorial

Now, divide \(_{7}P_{3}\) by 3!. Hence, by inserting the calculated values: \(\frac{_{7}P_{3}}{3!} = \frac{2100}{6} = 350\)
04

Calculate Combination

Next, calculate the combination \(_{7}C_{3}\) using the formula \(\frac{n!}{r!(n-r)!}\). Here, n=7 and r=3. Thus, \(_{7}C_{3} = \frac{7!}{3!(7-3)!}\). After calculation, we find that \(_{7}C_{3} = 35\).
05

Subtraction

The final step is to subtract the calculated combination from the division result of the permutation and factorial. That means \(350 - 35 = 315\).

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