44 CHAPTER 1 Sets, Proof Templates, and Induction
- The language department wanted to know how many of the 2000 students at the uni-
versity were not studying a language. Class rosters showed the number of students
studying some combination of French, German, and Spanish, as recapped in the fol-
lowing table:
Language No. of Students
French 75
German 68
Spanish 199
French and German 32
French and Spanish 41
German and Spanish 11
French and German and Spanish 7How many students were not studying a language?
How many integers between 1 and 250 are divisible by 3 or 5?
In the game of tic-tac-toe, every game ends with one player winning or with a draw.
In a tic-tac-toe tournament, the players merely count the number of times they win
or draw. The match winner is the player with the larger total. If a match between two
players A and B consists of 25 games, player A has a score of 19, and player B has a
score of 23, how many draws were there?
There are (^76) students enrolled in Anth229, Intermediate Anthropology. Each of these
students is also required to enroll in either one or both of Bio1313, Physiology, and
Engl218, Victorian Poets. Of these 76 students, there are 35 in Bio1313 and 49 in
Engl218. How many students are enrolled in all three classes?
The enrollment for the four courses Bio1212, Polil 15, Econ313, and Fina215 is 108,
203, 315, and 212, respectively. No student is in all four of these courses. No student is
in the three courses Biology 212, Fina215, and Poli 115. No student takes Econ313 and
Fina215 in the same semester. Polil 15 and Fina215 are not allowed in the same term.
There are 39 students in both Bio1212 and Poli 115, and 48 students in both Polil115
and Econ313 as well as in the two courses Bio1212 and Econ313. Bio1212, Poli115,
and Econ313 have a common enrollment of 73. Bio1212 and Fina215 have a common
enrollment of 67. How many different students are enrolled in these four courses?
How many numbers between 1 and 1000 are not divisible by 3, 7, or 9?
How many integers between 500 and 10,000 are divisible by 5 or 7?
(a) How many numbers between 1 and 70,000,000, including both 1 and 70,000,000,
are divisible by 2, 5, or 7?
(b) How many numbers between 1 and 6,000,000, including both 1 and 6,000,000,
are divisible by 4, 5, or 6?
Determine how many numbers between 1 and 21,000,000,000, including 1 and
21,000,000,000, are divisible by 2, 3, 5, or 7.
How many numbers between 1 and 21,000,000, including both 1 and 21,000,000, are
divisible by 2, 3, or 5 but not by 7?
Find the number of integers between 1 and 1000, including both 1 and 1000, that are
not divisible by any of 5, 6, or 8.