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 coe�cient of x3y6z12 in the expansion of (x + 2y2 + 4z3)10?
To answer these questions, we will break them down one by one and explain how to get the answer for each question.
1. How many students are taking:
(a) English and History but not Math?
To find the number of students taking English and History but not Math, we first find the number of students taking English and History (33). Then, we subtract the number of students taking all three subjects (13) since they are included in both English and History counts. Therefore, 33 - 13 = 20 students are taking English and History but not Math.
(b) Neither English, History, nor Math?
To find the number of students taking neither English, History, nor Math, we subtract the total number of students taking any of the three subjects from the total number of students. There are 110 students in total, and the sum of students taking English, History, and Math is 75 + 52 + 50 = 177. So, 110 - 177 = -67. Since it doesn't make sense to have negative students, the answer is 0.
(c) Math, but neither English nor History?
To find the number of students taking Math but not English or History, we start with the number of students taking Math (50). Then, we subtract the number of students taking all three subjects (13) since they are included in the Math count. Therefore, 50 - 13 = 37 students are taking Math but not English or History.
(d) English, but not History?
To find the number of students taking English but not History, we start with the number of students taking English (75). Then, we subtract the number of students taking all three subjects (13) since they are included in the English count. Therefore, 75 - 13 = 62 students are taking English but not History.
(e) Only one of the three subjects?
To find the number of students taking only one of the three subjects, we add the number of students taking English only, History only, and Math only. The number of students taking English only can be found by subtracting the number of students taking English and History (33) and the number of students taking all three subjects (13) from the total number of students taking English (75). Similarly, we find the numbers for History only (39) and Math only (27). Adding these numbers, 33 + 39 + 27 = 99 students are taking only one of the three subjects.
(f) Exactly two of the three subjects?
To find the number of students taking exactly two of the three subjects, we add the number of students taking two subjects (English and History, English and Math, or History and Math) excluding those taking all three subjects (13). Therefore, 33 + 30 + 22 - 13 = 72 students are taking exactly two of the three subjects.
Now let's move on to the next set of questions.