+18 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'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!).
+26367 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)
