A total of 36 students plan to take at least one of the courses Discrete Mathematics, Algebra and Calculus during the coming semester. Of these 36 students, it is known that 23 students plan to take Discrete Mathematics, 19 students plan to take Algebra, 18 students plan to take Calculus, 7 students plan to take Discrete Mathematics and Algebra, 9 students plan to take Discrete Mathematics and Calculus and 11 students plan to take Algebra and Calculus. (a) How many students plan to take all three courses. b) How many plan to take exactly one of the courses
c) How many plan to take exactly two of the courses

(a)5

(b)30

(c)12

Step-by-step explanation:

A total of 36 students plan to take at least one of the courses Discrete Mathematics(D), Algebra(A) and Calculus(C)

Let x be the number who take all three courses.

From the Diagram

Number who takes Discrete Mathematics (D) Only =23-(7-x+x+9-x) =23-(16-x)=23-16+x = 7+x

Number who takes Algebra (A) Only =19-(7-x+x+11-x) =23-(18-x)=23-18+x = 5+x

Number who takes Calculus (C) Only =18-(9-x+x+11-x) =23-(20-x)=23-20+x = 3+x

Adding up all 7 regions of the Venn Diagram should give 36

(7+x)+(7-x)+x+(9-x)+(5+x)+(11-x)+(3+x)=36

7+7+9+5+3+x=36

31+x=36

x=36-31=5

(a) 5 Students takes all three courses

(b) Number of Students who plan to take exactly one course

Number who takes Discrete Mathematics (D) Only+Number who takes Algebra (A) Only+ Number who takes Calculus (C) Only

=(7+x)+(5+x)+(3+x)

=15+3x

=15+3(5) =15+15=30

(c)Number of students who plan to take exactly two of the courses

n(A∩C)+n(A∩D)+n(C∩D)=(11-x)+(7-x)+(9-x)

=27-3x = 27-3(5)=27-15=12

when rotating a figure 180 degrees, imagine that you are able to take the figure and turn it clockwise or counterclockwise around a center point. to rotate a figure 180 degrees, you will need to apply the rule (x, y) → (-x, -y). start by using a coordinate grid with coordinates for each vertex of the figure. the center point of the coordinate grid is located at (0, 0), which is what you will rotate the figure around. write down the original coordinates of the shape you are going to rotate. then, apply the rule. for example, coordinate (1, 2) becomes (-1, -2). another example would be, coordinate (-4, -2) becoming (4, 2). after you change each original coordinate to the rotated coordinate, you will draw your new figure.

step-by-step explanation: