In a survey of 269 college students, it is found that
69 like brussels sprouts,
90 like broccoli,
59 like cauliflower,
28 like both brussels sprouts and broccoli,
20 like both brussels sprouts and cauliflower,
24 like both broccoli and cauliflower, and
10 of the students like all three vegetables.

a) how many of the 269 college students do not like any of these three vegetables?

b) how many like broccoli only?

c) how many like broccoli and cauliflower but not brussels sprouts?

d) how many like neither brussels sprouts nor cauliflower?

a) 83, b) 28, c) 14, d) 28.

Step-by-step explanation:

Since we have given that

n(B) = 69

n(Br)=90

n(C)=59

n(B∩Br)=28

n(B∩C)=20

n(Br∩C)=24

n(B∩Br∩C)=10

a) How many of the 269 college students do not like any of these three vegetables?

n(B∪Br∪C)=n(B)+n(Br)+n(C)-n(B∩Br)-n(B∩C)-n(Br∩C)+n(B∩Br∩C)

n(B∪Br∪C)=

So, n(B∪Br∪C)'=269-n(B∪Br∪C)=269-156=83

b) How many like broccoli only?

n(only Br)=n(Br) -(n(B∩Br)+n(Br∩C)+n(B∩Br∩C))

n(only Br)=

c) How many like broccoli AND cauliflower but not Brussels sprouts?

n(Br∩C-B)=n(Br∩C)-n(B∩Br∩C)

n(Br∩C-B)=

d) How many like neither Brussels sprouts nor cauliflower?

n(B'∪C')=n(only Br)= 28

Hence, a) 83, b) 28, c) 14, d) 28.

Total number of boys at the concert=100

Step-by-step explanation:

The total number of boys and girls at the concert can be expressed as follows;

Total number of boys (b)=(ratio of boys/total ratio)×total number of boys and girls (t)

where;

Total number of boys=b

ratio of boys=2

total ratio=(2+7)=9

total number of boys and girls=t

replacing;

b=(2/9)×t...equation 1

The total number of girls (g)=Total number of boys (b)+250

g=b+250...equation 2

The value of b in equation 2 is replaced by the value of b in equation 1;

g=(2/9)t+250

And the total number can be expressed as;

Total number (t)=Total number of boys (b)+total number of girls (g)

t=b+g

where;

b=(2/9)t

g=(2/9)t+250

replacing;

t=(2/9)t+(2/9)t+250

t=(4/9)t+250

t-(4/9)t=250

(5/9)t=250

t=250×9/5

t=450

The total number of boys (b)=(2/9)t

replace t with 450

b=(2/9)×450

b=100

Total number of boys at the concert=100