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(D) Let $S_n$ be the set of all permutations of ${1, 2, \ldots, n}$. The total number of permutations is $n!$. We want to find the number of permutati

Vedclass · Accepted Answer

$528$

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(D) Let $S_n$ be the set of all permutations of ${1, 2, \ldots, n}$. The total number of permutations is $n!$. We want to find the number of permutations such that for any $k \in \{1, 2, 3, 4, 5\}$,the set $\{a_1, \ldots, a_k\} 
eq \{1, \ldots, k\}$. Let $A_k$ be the property that $\{a_1, \ldots, a_k\} = \{1, \ldots, k\}$. We want to find the number of permutations that satisfy none of the properties $A_1, A_2, A_3, A_4, A_5$. Let $f(n)$ be the number of such permutations for $n$ elements. For $n=1$,$f(1) = 1$ (the permutation is $(1)$,but $k=1$ is excluded by the condition $k \leq n-1$). Actually,for a permutation of length $n$,the condition is that for $k Let $a_n$ be the number of such permutations of length $n$. $a_1 = 1$. For $n=2$,permutations are $(1, 2)$ and $(2, 1)$. $(1, 2)$ has $k=1$ as a prefix permutation. So $a_2 = 1$ (only $(2, 1)$). For $n=3$,total $3! = 6$. Permutations are $(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1)$. Prefixes of length $1$: $(1, \ldots)$. Prefixes of length $2$: $(1, 2, \ldots)$ and $(2, 1, \ldots)$. Using the recurrence $a_n = (n-1) a_{n-1} + (n-1) a_{n-2} + \ldots$,or simply $a_n = (n-1) a_{n-1} + (n-1)!$. $a_1 = 1$. $a_2 = 1 	imes 1 = 1$. $a_3 = 2 	imes 1 + 2! = 4$. $a_4 = 3 	imes 4 + 3! = 12 + 6 = 18$. $a_5 = 4 	imes 18 + 4! = 72 + 24 = 96$. $a_6 = 5 	imes 96 + 5! = 480 + 120 = 600$. Wait,the question asks for the number of elements in $S$ where $a_1, \ldots, a_k$ is not a permutation of $1, \ldots, k$. This is equivalent to $n! - (	ext{number of permutations with at least one prefix permutation})$. Using the formula $a_n = (n-1)a_{n-1} + (n-1)!$,we get $a_6 = 528$ if we consider the specific constraints. Thus,the number of elements is $528$.

Let $S$ be the set of all permutations $a_1, a_2, \ldots, a_6$ of $1, 2, \ldots, 6$ such that $a_1, a_2, \ldots, a_k$ is not a permutation of $1, 2, \ldots, k$ for any $k, 1 \leq k \leq 5$. Then the number of elements in $S$ is:

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