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A class of college freshmen is composed of 110 students. Among these students,
75 are taking English, 52 are taking history, 50 are taking math, 33 are taking English
and history, 30 are taking English and Math, 22 are taking History and Math, and 13
are taking all three subjects. How many students are taking:
(a) English and History but not Math?
(b) neither English, History, nor Math?
(c) Math, but neither English nor History?
(d) English, but not History?
(e) only one of the three subjects?
(f) exactly two of the three subjects?
2. 2 pts. How many di erent signals, each consisting of 8
ags hung in a vertical line,
can be formed from a set of 4 indistinguishable red
ags, 3 indistinguishable white

ags, and a blue
ag?
3. 2 pts. In how many ways can 3 boys and 2 girls sit in a row? In a round table?
4. 4 pts. In how many ways can 3 Americans, 4 Frenchmen, 4 Italians, and 2 Chinese
be seated in a row so that those of the same nationality sit together? What if they
were to be seated in a round table?
5. 2 pts. Expand and simplify: (x2 􀀀 2y)6.
6. 2 pts. How many solutions are there to the equation n1 + n2 + n3 + n4 = 21 where
n1  2; n2  3; n3  4, and n4  5?
7. 2 pts. What is the coecient of x3y6z12 in the expansion of (x + 2y2 + 4z3)10?

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