(related to Problem: The Wrong Hats)
The number of different ways in which eight persons, with eight hats, can each take the wrong hat, is $14,833.$
Here are the successive solutions for any number of persons from one to eight:— $1 = 0$ $2 = 1$ $3 = 2$ $4 = 9$ $5 = 44$ $6 = 265$ $7 = 1,854$ $8 = 14,833$
To get these numbers, multiply successively by $2,$ $3,$ $4,$ $5,$ etc. When the multiplier is even, add $1;$ when odd, deduct $1.$ Thus, $3 \times 1- 1 = 2,$ $4 \times 2 + 1 = 9;$ $5 \times 9- 1 = 44;$ and so on. Or you can multiply the sum of the number of ways for $n-1$ and $n-2$ persons by $n-1,$ and so get the solution for $n$ persons. Thus, $4(2 + 9) = 44;$ $5(9 + 44) = 265;$ and so on.
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