The complement event of "at least one man is assigned to the swing shift" is
the event that all 3 women are assigned to the swing shift.
That's 

  the number of ways to have all 3 women on the swing shift      1       1
------------------------------------------------------------- = ---- = -----
the number of ways to have any 3 of the 10 on the swing shift   10C3    120

So 1/120 is the probability that all 3 women will be on the swing shift.

Therefore 1 - 1/120 or 119/120 is the probability that there will be at least
one man on the swing shift.

[It doesn't matter about the other two shifts.  We can ignore them. We are
considering each of the 10 people either to be assigned to the swing shift or
elsewhere.]

Edwin