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# A permutation of the numbers (1,2,3,...,n) is a rearrangement of the numbers in which each number appears exactly once.

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A permutation of the numbers (1,2,3,...,n) is a rearrangement of the numbers in which each number appears exactly once. For example, (2,5,1,4,3) is a permutation of (1,2,3,4,5). Let \pi = (x_1,x_2,x_3,---,x_n) be a permutation of the numbers (1,2,3,....,n). A fixed point of \pi is an integer k(1 ≤ k ≤ n) such that x_k=k. For example, 4 is a fixed point of the permutation $(2,5,1,4,3). How many permutations of (1,2,3,4,5,6,7) have at least one even fixed point? Nov 3, 2019 ### 2+0 Answers #1 0 I'm not sure if I understand your question fully!. But, I will give it a try. My undestanding is that you want to know how many pemutations are there where 2 (even number) is in the 2nd position from the left, and 4 (even number) is in the 4th position from the left and 6 (even number) in the 6th position from the left. If that is the case, then: 7! =5,040 permutations. Each of the 7 numbers appears:5,040 / 7 =720 times in EACH of the 7 positions, from left to right. So, 2 will appear in the 2nd position from the left 720 times. So will 4 appear in the 4th position from the left 720 times as well. And so will 6 appear in the 6th position from the left 720 times. Therefore, all three even numbers(2, 4, 6) will appear in their repective positions in: 3 x 720 =2,160 permutations in total (if I understood your question!). Nov 3, 2019 #2 +24365 +1 A permutation of the numbers (1,2,3,...,n) is a rearrangement of the numbers in which each number appears exactly once. For example, (2,5,1,4,3) is a permutation of (1,2,3,4,5). Let \pi = (x_1,x_2,x_3,---,x_n) be a permutation of the numbers (1,2,3,....,n). A fixed point of \pi is an integer k(1 ≤ k ≤ n) such that x_k=k. For example, 4 is a fixed point of the permutation$(2,5,1,4,3). How many permutations of (1,2,3,4,5,6,7) have at least one even fixed point?

I assume 1824 permutations of (1,2,3,4,5,6,7) have at least one even fixed point.

1.) 1234567   3 (even fixed points)
2.) 1234576   2 (even fixed points)
3.) 1234657   2 (even fixed points)
4.) 1234675   2 (even fixed points)
5.) 1234765   3 (even fixed points)
6.) 1234756   2 (even fixed points)
7.) 1235467   2 (even fixed points)
8.) 1235476   1 (even fixed points)
9.) 1235647   1 (even fixed points)
10.) 1235674   1 (even fixed points)
11.) 1235764   2 (even fixed points)
12.) 1235746   1 (even fixed points)
$$\cdots$$
998.) 4731265   1 (even fixed points)
999.) 4735162   1 (even fixed points)
1000.) 4735261   1 (even fixed points)
1001.) 4732561   1 (even fixed points)
1002.) 4732165   1 (even fixed points)
1003.) 4713562   1 (even fixed points)
1004.) 4713265   1 (even fixed points)
1005.) 4715362   1 (even fixed points)
1006.) 4715263   1 (even fixed points)
$$\cdots$$
1410.) 6217453   1 (even fixed points)
1411.) 6274513   2 (even fixed points)
1412.) 6274531   2 (even fixed points)
1413.) 6274153   2 (even fixed points)
1414.) 6274135   2 (even fixed points)
1415.) 6274315   2 (even fixed points)
1416.) 6274351   2 (even fixed points)
1417.) 6275413   1 (even fixed points)
1418.) 6275431   1 (even fixed points)
1419.) 6275143   1 (even fixed points)
$$\cdots$$
1626.) 7261453   1 (even fixed points)
1627.) 7214563   3 (even fixed points)
1628.) 7214536   2 (even fixed points)
1629.) 7214653   2 (even fixed points)
1630.) 7214635   2 (even fixed points)
1631.) 7214365   3 (even fixed points)
1632.) 7214356   2 (even fixed points)
$$\cdots$$
1812.) 7164325   1 (even fixed points)
1813.) 7164235   1 (even fixed points)
1814.) 7164253   1 (even fixed points)
1815.) 7124563   2 (even fixed points)
1816.) 7124536   1 (even fixed points)
1817.) 7124653   1 (even fixed points)
1818.) 7124635   1 (even fixed points)
1819.) 7124365   2 (even fixed points)
1820.) 7124356   1 (even fixed points)
1821.) 7125463   1 (even fixed points)
1822.) 7125364   1 (even fixed points)
1823.) 7123564   1 (even fixed points)
1824.) 7123465   1 (even fixed points)

Nov 4, 2019