Square dance with 10 people in a line 4 amazons standing in the wrong place Ask how many wrong arrangements there are

10 people in a column, 4 standing in the wrong place.

This question can be discussed in two steps.

In the first step, find the number of scenarios in which all 4 amazons are standing in the wrong place.

This is a "Bernoulli's Misfiled Envelope Problem", and there are recursive formulas for this. For 4 people, the answer can also be found by enumeration, and a **** is 9 scenarios. See the diagram below.

In the second step, find the number of combinations of 10 choices of 4.

10!/4!/(10-4)! = 210 combinations.

The two steps stack up, and one *** is 9*210 = 1890 wrong permutations.