Answer to Question #101168 in Algebra for Barbara Baaba Godwyll

Question

1. A group of 46 students were required to choose from Agric(A), Biology(B) and Chemistry (C) subjects. The following were their preferences: (A n B) n C' = 7,
(A n B) = 11, (A' n B n C') = 4, n(A) = 16, n(B) = 18, n(C) = 17 and (A u B u C)' = x.

a. Draw a venn diagram to illustrate the information.
b. Find the value of x.
c. Find the number of students who preferred
i. Only one of the subjects
ii. At most one of the subjects
iii. At least two of the subjects

Answer 1

$A \cap B \cap C´=c=7$

$A \cap B=a+c=11 \Rightarrow a=11-c=11-7=4$

$A’ \cap B \cap C’=f=4$

$A\cap B’ \cap C’=g=5$ (this condition was assumed in order to solve a question)

$B=a+b+c+f=18 \; \Rightarrow b=18-a-c-f=18-4-7-4=3$

$A=a+c+g+d=16 \;\Rightarrow d=16-a-c-g=16-4-7-5=0$

$C=a+b+d+e=17 \;\Rightarrow e=17-a-b-d=17-4-3-0=10$

$A\cup B \cup C=a+b+c+d+e+f+g=4+3+7+0+10+4+5=33$

$x=33, \quad x’=46-33=13$ students choose no subject

1) Only one of the subject preferred: $y=e+f+g=10+4+5=19$ students

2) At most one subject preferred: $y+x’=19+13=32$ students

3) At least two subjects preferred: $a+b+c+d=7+3+4+0=14$ students