I solved it once without derangements... think of this name-drawing as a permutation where each spot represents a person... and then those spots get filled in with their names (ie 1 through 6) in some order.... 123456 143265 654312 etc... being in the correct spot represents drawing your own name.

- Number of permutations where all 6 draw their name: 1

- Number of permutations where exactly five draw their own name: 0 (it's impossible)

- Number of permutations where exactly four draw their own name: 6C4*1 = 15

- Number of permutations where exactly three draw their own name: 6C3*2=40 (6C3 ways to choose which three draw their own names and 2 ways for the remaining people to not draw their names)

- Number of permutations where exactly two draw their own name: 6C2*Q where Q=the number of ways for four people to not draw their own name.

*To get Q you have to recursively use the previous steps...* think permutations like 1243 1234 etc.

*permutations (length 4) where all 4 draw their own name: 1 *

*permutations where exactly 3 draw their own name: 0 *

*permutations where exactly 2 draw their own name: 4C2*1=6*

*permutations where exactly 1 draws their own name: 4C1*2=*8

*permutations where none draw their own name: (4x3x2x1)-(1+6+8)=9*

*therefore Q=9 *

6C2*Q=6C2*9= 135 and therefore there are 135 permutations (length 6) where exactly 2 people draw their own name.

- number of permutations where exactly 1 person draws their own name: 6C1*L where L is the number of ways for 5 people to not draw their own names

*permutations (length 5) where all 5 draw their own name: 1*

permutations where exactly 4 draw their own name: 0

permutations where exactly 3 draw their own name: 5C3*1= 10

permutations where exactly 2 draw their own name: 5C2*2= 20

permutations where exactly 1 draws their own name: 5C1*Q=5C1*9= 45

permutations where none draw their own name: (5x4x3x2x1)-(1+10+20+45)=44

therefore L=44

6C1*L=6C1*44= 264 and therefore there are 264 permutations (length 6) where exactly 1 person draws their own name.

Therefore, the probability of no one drawing their own name in a group of six people is:

1-P(1 or 2 or 3 or 4 or 5 or 6 people draw their own name)

=1- (1+15+40+135+264)/(6x5x4x3x2x1)

=0.36805555555