Problem source

There are 28 colleges in Cambridge, of which two (New Hall and Newnham)
are for women only; the others admit both men and women. Seven women,
Anya, Betty, Celia, Doreen, Emily, Fariza and Georgina, are all applying
to Cambridge. Each has picked three colleges at random to enter on
her application form. 

\begin{questionparts}
\item  What is the probability that Anya's first choice college is
single-sex?
\item What is the probability that Betty has picked Newnham?
\item What is the probability that Celia has picked at least one
single-sex college?
\item Doreen's first choice is Newnham. What is the probability that
one of her other two choices is New Hall?
\item Emily has picked Newnham. What is the probability that she
has also picked New Hall?
\item Fariza's first choice college is single-sex. What is the probability
that she has also chosen the other single-sex college?
\item One of Georgina's choices is a single-sex college. What is
the probability that she has also picked the other single-sex college?
\end{questionparts}

Solution source

\begin{questionparts}
\item $\frac{2}{28} = \frac{1}{14}$ 
\item $1-\frac{27}{28}\frac{26}{27}\frac{25}{26} = \frac{3}{28}$
\item $1 - \frac{26}{28}\frac{25}{27}\frac{24}{26} = \frac{13}{63}$
\item $1-\frac{26}{27}\frac{25}{26} = \frac{2}{27}$
\item $\frac{1}{27}$
\item There are $\binom{2}{1} \binom{26}{2} + \binom{2}{2}\binom{26}{1}$ ways to choose at least one single sex college and $ \binom{2}{2}\binom{26}{1}$ ways to choose both, therefore

\begin{align*}
P &= \frac{ \binom{2}{2}\binom{26}{1}}{\binom21 \binom{26}2+ \binom{2}{2}\binom{26}{1}} \\
&= \frac{26}{2\cdot \frac{26\cdot25}{2}+26 }\\
&= \frac{1}{25+1} = \frac{1}{26}
\end{align*}
\end{questionparts}