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Chapter 8: Problem 4

How many permutations of \(1,2,3,4,5,6,7\) are not derangements?

Short Answer

Expert verified
The number of permutations of \(1,2,3,4,5,6,7\) which are not derangements is calculated by subtracting the total number of derangements from the total number of permutations. This is found by applying the relevant permutations and derangements formulas.

Step by step solution

01

Calculate Total Permutations

The total number of permutations for a set of 'n' objects is given by the formula \(n!\) (n factorial). For the given set of 7 objects, this will be \(7!\). Calculate this value.
02

Calculate Total Derangements

The total number of derangements for a set of 'n' objects can be calculated using the formula \[ !n = n! \left( \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + .. + (-1)^n \frac{1}{n!} \right) \]. Apply this formula for n=7 to calculate total derangements.
03

Calculate Permutations which are not Derangements

The permutations which are not derangements can be found by subtracting the total number of derangements from the total number of permutations. Using the values found in Step 1 and Step 2, perform this subtraction to find the answer.

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Most popular questions from this chapter

In how many ways can three \(x\) 's, three \(y\) 's, and three \(z\) 's be arranged so that no consecutive triple of the same letter appears?

In how many ways can one arrange the letters in CORRESPONDENTS so that (a) there is no pair of consecutive identical letters? (b) there are exactly two pairs of consecutive identical letters? (c) there are at least three pairs of consecutive identical etters?

a) Given \(n\) distinct objects, in how many ways can we select \(r\) of these objects so that each selection includes some particular \(m\) of the \(n\) objects? (Here \(m \leq r \leq n\).) b) Using the principle of inclusion and exclusion, prove that for \(m \leq r \leq n\), $$ \left(\begin{array}{c} n-m \\ n-r \end{array}\right)=\sum_{i=0}^{m}(-1)^{(}\left(\begin{array}{c} m \\ i \end{array}\right)\left(\begin{array}{c} n-i \\ r \end{array}\right) $$

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