5.6.9 (Refer to figure 3.) How many routes from A to B pass through the
point C?

Hint: Such a route can be written down in two stages -- choose a route from
A to C, and then a route from C to B. Use the multiplication principle.

-----

5.6.11 In how many ways can a committee of 5 senators be selected from the
100 members of the U.S. Senate so that no two committee members come from
the same state?

Hint: Each state has 2 senators. Select a committee in 2 stages -- first
select the 5 states involved, and then select which of the two senators is
chosen in each case. (The second stage is really 5 different selections.)

-----

5.6.36 In how many ways can a residence director assign 6 students to
four dormitory rooms if there are two double rooms and 1 single room and if
two of the students cannot be placed together?


Hint: Call the unfriendly students A and B, and the others D-G. Do the
assignment in the following stages: First assign A to one of the rooms and
then B to one of the rooms other than the one where A was put. That leaves
4 "slots" for the rest of the students, who can be distributed among them
arbitrarily.