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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
 #1
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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
avatar+23516 
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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?

 

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)

 

laugh

 Nov 4, 2019

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