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Chapter 11: Problem 36

Use the formula for \({ }_{n} P_{r}\) to evaluate each expression. \({ }_{10} P_{4}\)

Short Answer

Expert verified
The total number of possible arrangements of 4 items taken out from a total of 10 items is 7.

Step by step solution

01

Understand the Problem

In this problem, we are going to find out the total possible number of arrangements for \(4\) items taken out from a total of \(10\) items. Here, the order of the items matters.
02

Recall the Formula

The permutation formula for 'n' items taken 'r' at a time is given by \({ }_{n} P_{r} = \frac{n!}{(n-r)!}\), where '!' represents a factorial.
03

Apply the Formula

Using the formula, we plug in \(n = 10\) and \(r = 4\). So we get \({ }_{10} P_{4} = \frac{10!}{(10-4)!}\).
04

Calculate Factorials

Factorial means to multiply a series of descending natural numbers. We first compute \(10!\) which equals to \(10 脳 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1\) and \(6!\) which equals to \(6 脳 5 脳 4 脳 3 脳 2 脳 1\). But since both !'s go down to \(1\) and the factorials are in division, we only have to calculate down to \(6!\) for both. So we calculate \(10 脳 9 脳 8 脳 7\) for the \(10!\) which equals \(5040\) and divide by \(6!\) which equals \(720\).
05

Complete the Calculation

We divide \(5040\) by \(720\) and get \(7\). So, \({ }_{10} P_{4} = 7\).

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