A = arts club
B = academic club
C = music club
This is what the Venn Diagram looks like
We have these 8 distinct regions marked in red
a = number in arts club only
b = number in arts & academics, but not music
c = number in academics club only
d = number in arts & music, but not academic
e = number of students in all 3 clubs
f = number in academics & music, but not arts
g = number in music club only
h = number of students in neither of the 3 clubs mentioned
I'll break the given info into these facts
Fact [1]: 6 of the students belong to only the Arts Club.
Fact [2]: 5 of the students belong to all 3 clubs.
Fact [3]: 2 of the students belong to the Academic and Arts clubs but not to the Music club.
Fact [4]: 15 of the students belong to the Arts Club.
Fact [5]: 2of the students belong only to the Academic Club.
Fact [6]: 3 of the students belong only to the Music Club.
Based on those facts we can then determine the following:
a = 6 .... from Fact [1]
b = 2 .... from Fact [3]
c = 2 .... from Fact [5]
d = unknown for now
e = 5 .... from Fact [2]
f = unknown for now
g = 3 .... from Fact [6]
h = 0 ... because every student mentioned belongs to at least one of the three clubs
We've used nearly every fact, except fact 4, to determine values a,b,c,e,g,and h.
Use Fact [4] to form and solve this equation
a+b+d+e = 15 students in the arts club (circle A)
6+2+d+5 = 15
13+d = 15
d = 15-13
d = 2
Now we'll use the fact that there are 30 students total, each of which belongs to one or more clubs.
a+b+c+d+e+f+g+h = 30
6+2+2+2+5+f+3+0 = 30
20+f = 30
f = 30-20
f = 10
Here's a summary of all the values
a = 6
b = 2
c = 2
d = 2
e = 5
f = 10
g = 3
h = 0
and here's what the Venn diagram looks like now after replacing the letters with their corresponding numbers.
We're now told to "Find the number of students who belong to at least two clubs".
There are two ways to do this:
Method 1)
Add up the values b, d, e, f as they represent students who are in 2 clubs or more.
b+d+e+f = 2+2+5+10 = 19
There are 19 students who are in 2 clubs or more.
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Method 2)
As a slight detour, let's find out how many students are in exactly one club only.
artsOnly + academicsOnly + musicOnly = a + c + g = 6+2+3 = 11
There are 11 students who have signed up for exactly one club.
There are 30 students total, so that must mean 30-11 = 19 students are in at least two clubs.
The events "in exactly one club" and "in at least two clubs" are complementary events. One or the other must happen, but not both at the same time.
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Final Answer: 19 students who belong to at least two clubs.