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Chapter 6: Problem 22

How many six-letter sequences are possible that use the letters \(\mathrm{q}, \mathrm{u}, \mathrm{a}, \mathrm{k}, \mathrm{e}, \mathrm{s}\) once each?

Short Answer

Expert verified
There are 720 possible six-letter sequences using the letters $\mathrm{q}, \mathrm{u}, \mathrm{a}, \mathrm{k}, \mathrm{e}, \mathrm{s}$ once each.

Step by step solution

01

Identify what needs to be calculated

We need to find the number of ways to arrange 6 different letters in a sequence.
02

Use the permutation formula

In this situation, we are using all of the available letters, and the order in which they're placed matters. So, we can use the formula for permutation of n distinct objects taken r at a time, given by: \(P(n,r) = \frac{n!}{(n-r)!}\) In our case, n = 6 (number of letters) and r = 6 (number of positions for the sequence).
03

Calculate the factorial and substitution

First, find the value of \(n!\) and \((n-r)!\) We have \(n! = 6! = 6 脳 5 脳 4 脳 3 脳 2 脳 1 = 720\) Since \(r = n\), we have \((n-r)! = (6-6)! = 0! = 1\) Now, substitute the values into the formula: \(P(6,6) = \frac{6!}{(6-6)!} = \frac{720}{1}\)
04

Calculate the number of sequences

Calculating the result: \(P(6,6) = \frac{720}{1} = 720\) Therefore, there are 720 possible six-letter sequences using these letters once each.

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