Chapter 7, Problem 25RE

On the day before midterm break, 20 of the 32 students enrolled in a psychology course decide to skip class, Asked in how many ways can this be accomplished if Tim and Nicole refuse to skip together, student X and Y come up with apparently different answers.

X argues this way. There are $\left({}_{19}^{30}\right)$ ways in which Tim skips, but not Nichole, and another $\left({}_{19}^{30}\right)$ in which Nichole skips, but not Tim. Adding the $\left({}_{20}^{30}\right)$ ways in which neither skips gives $2\left({}_{19}^{30}\right)+\left({}_{20}^{30}\right)$.

Y says that there are $\left({}_{18}^{30}\right)$ ways in which both Tim and Nichole skip, so the answer is the total number of possibilities less this number, that is, $\left({}_{20}^{30}\right)-\left({}_{18}^{30}\right)$.

Who is right?