I am taking Introductory Discrete Mathematics?

Question

I have this problem that I can't get through. It states: In a survey of 4 year colleges and universities it was found that 356 offered a liberal arts degree, 293 offered a computer engineering degree and 285 offered a nursing degree, 193 offered a liberal arts & computer engineering degree, 200 offered a liberal arts and nursing degree, 139 offered a computer engineering and nursing degree, 68 offered a liberal arts, computer engineering and nursing degree and 26 offered none of these degrees.

It wants you to draw a Venn diagram and answer these questions:

a. How many 4 year colleges and universities were surveyed?

b. How many offered liberal arts and a nursing degree but not a computer engineering degree?

c. How many offered a computer engineering degree but neither a liberal arts or a nursing degree?

d. How many offered exactly 2 of these degrees?

e. How many offered at least one of these degrees?

I am so confused on these one. Please help. I filled everything in but I am coming up minuses for one of them. And it doesn't add up to the 1st three degrees. I am off somewhere and have been working on these problem for about 6 hours.

Answer 1

The first place to start is the centre of the diagram. its been given that 68 offered all 3 so place a 68 in the middle of your diagram.

The number in nursing and arts is 200. this includes all the 68 schools that offered all 3 as well so you minus 68 from 200 to get those schools that only offered those 2 things. the number that offers nursing and arts is 132.

do the same thing for the other 2 double subject university's you should get computing and nursing = 139 - 68 = 71 and computing and arts = 193 - 68 = 125. draw these numbers onto your diagram.

then add up all the circles and minus the total in each circle from the total number of schools that offer that subject. for example

nursing = 285 - (132+68+71) = 14

computing = 293 - (125+68+71) = 29

arts = 356 - (125+132+68)=31

outside your venn diagram all you need to do is draw in the 26 to represent the schools with no subjects

a)solve by adding all the numbers in all sections

14+132+68+71+31+125+29+26 = 496

b)just read this off your table its the section in the nursing and arts circle but not in the computing. the answer should be 132

c)in the computing circle only =  29

d) exactly 2 add up all the inner circles except the 68(because thats 3 subjects) 132+71+125 = 328

E) at least one of the degrees is everybody minus 26 for the schools that didnt compete.

ie 496 - 26 = 470

thats the method hope those answers are right if not ive done something wrong on a calculator