Answer
$\frac{19}{30}$
Work Step by Step
Total ways to insert paychecks is $5 !=120$ ways
Now count the number of ways we can have 5,4,3,2,1,0 correct envelopes :
5 correct: 1 way
4 correct: not possible, because the fifth will be correct as well.
3 correct $:{ }_{5} C_{3} \cdot 1=10$ ways
( once the three envelopes that will contain the correct paychecks have been chosen, there is only one way to insert the paychecks so that the other two are wrong)
2 correct $:{ }_{5} C_{2} \cdot 2 \cdot 1=20$ ways
( once the two envelopes that contain the correct paychecks are chosen, for the three that remain, there are two ways to fill the next envelope incorrectly, then only one incorrect way to insert the remaining paychecks)
1 correct: $\quad 5 \cdot 3 \cdot 3 \cdot 1=45$ ways
(five ways to choose which envelope has the correct paycheck, of the remaining four,three ways to fill the next envelope incorrectly, then three ways to fill the envelope whose correct paycheck was placed in another envelope, and only one way to fill the remaining two envelopesso that both are incorrect)
0 correct: $120-1-10-20-45=44 $
Answer for part (b)
$\frac{45+20+10+1}{120} = \frac{19}{30}$
