# Problems on Derangements - Combinatorics [closed]

Determine the number of permutations of $1,2, \ldots ,8$ in which no even integer is in its natural position. Please solve this using concept of derangements.

## closed as off-topic by ml0105, choco_addicted, M. Vinay, colormegone, David KMay 28 '16 at 19:46

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• Well solve it then, whats the problem! – Alex May 28 '16 at 14:33
• @Alex my answer not matching. If we had to calculate $D_4$ – mathlover May 28 '16 at 14:34
• You didn't show any work, how am i supposed to know! – Alex May 28 '16 at 14:43
• @Ayush: The answer is not $D_4$: that’s just the number of permutations that derange the even numbers amongst themselves and leave the odd numbers fixed. The permutation that interchanges $1$ and $2$, $3$ and $4$, $5$ and $6$, and $7$ and $8$ leaves no even integer in its natural position and is not counted in your $D_4$. – Brian M. Scott May 28 '16 at 15:48
• What does "using concept of derangements" mean? Is it enough to use the same technique, i.e. Inclusion/Exclusion? – André Nicolas May 28 '16 at 16:07

Let $S$ be the set of all permutations of $\{1, 2, . . . , 8\}$. Let $A_1$ be the set of all permutations in $S$ which has $2$ in it natural position; $A_2$ be the set of all permutations in $S$ which has $4$ in it natural position; $A_3$ be the set of all permutations in $S$ which has $6$ in it natural position and $A_4$ be the set of all permutations in $S$ which has $8$ in it natural position. Then

$|A_1| = |A_2| = |A_3| = |A_4| = (8 − 1)!$,

$|A_1 ∩ A_2| = |A_1 ∩ A_3| = |A_1 ∩ A_4| = |A_2 ∩ A_3| = |A_2 ∩ A_4| = |A_3 ∩ A_4| = (8 − 2)!$,

$|A_1 ∩ A_2 ∩ A_3| = |A_1 ∩ A_2 ∩ A_4| = |A_1 ∩ A_3 ∩ A_4| = |A_2 ∩ A_3 ∩ A_4| = (8 − 3)!$,

$|A_1 ∩ A_2 ∩ A_3 ∩ A_4| = (8 − 4)!$. Then the set of permutations of $\{1, 2, . . . , 8\}$ in which no even integers are in their national positions is $\bar A_1 ∩ \bar A_2 ∩ \bar A_3 ∩ \bar A_4$ and the number of the elements is $8! − 4 · 7! + 6 · 6! − 4 · 5! + 4! = 24024.$ $\blacksquare$

(And also we can use this identity which is related to derangement : $n! = \Sigma _{k=0}^n \binom {n}{k}D_{n-k}= \Sigma _{k=0}^n \binom {n}{k}D_{k}.$

Proof: Let $S$ be the set of all permutations of $\{1, 2, . . . , n\}$. Let $A_k$ be the set of all permutations that $k$ integers are fixed at their positions. Then $|S|= n!$ and $|A_k| = \binom{n}{k}D_{n-k}$ . The identity follows from the disjoint union $S = \cup^{n}_{k=0} A_k.$)

The total number of permutations of 1-8 is $8!$. Of these $7!$ fix any one given even number, $6!$ fix two, $5!$ fix three and $4!$ fix all four. So by inclusion-exclusion the number fixing none is $8!-4\cdot7!+6\cdot6!-4\cdot5!+4!=24024$.