Total = (S U B U V) + (S U B U V)'


Total = S + B + V - (S ∩ B) - (S ∩ V) - (B ∩ V) + (S ∩ B ∩ V) + (S U B U V)'

Since each are playing at least 1 of the games, (S U B U V)' = 0


(S ∩ B) + (S ∩ V) + (B ∩ V) = S + B + V + (S ∩ B ∩ V) - Total = 50 + 45 + 50 + 15 - 100 = 60


Only (S ∩ B) = (S ∩ B) - (S ∩ B ∩ V)

Only (S ∩ V) = (S ∩ V) - (S ∩ B ∩ V)

Only (B ∩ V) = (B ∩ V) - (S ∩ B ∩ V)


Therefore those who play only 2 games = (S ∩ B) + (S ∩ V) + (B ∩ V) - 3 * (S ∩ B ∩ V) = 60 - (3 * 15) = 60 - 45 = 15


Option C

Arun Kumar
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ANSWER C 15
Or
the sortest way is
50+50+45=145
15 is calculated trice so 15*3=45
145-45=100
now the total number of students are 100-15 =85
now in 100 some students are recorded twice and some are recorded once
so if i subtracted 100-85 i will get only that the number that are recorded twice that is 15

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IMO C

(sum of students who play a particular sport) - (sum of students who play two games) - 2* (sum of students who play all three) = total
(50+45+50) - (X) - 2*15 = 100
145 - X - 30 = 100
=> X = 15

In a class, there are 100 students. 50 of them play soccer, 45 basketball and 50 play volleyball. Only 15 play all three games. Everyone plays at least one game. How many play only two games?

A. 5
B. 10
C. 15
D. 20
E. 25

X = play only two games

100 = 50 + 50 + 45 - X - 2*15
X = 15

Answer C

The equation for sets of three is helpful:

A + B + C - (exactly 2) - 2*(all three) = total

Soccer = 50. We'll plug that in for A
Basketball = 45. We'll plug that in for B
Volleyball = 50. We'll plug that in for C
Given that 15 students play all three sports
We just need to find the (exactly 2) part of the equation

Equation is: 50 + 45 + 50 - (exactly 2) - 2(15) = 100 total students
145 - exactly 2 - 30 = 100
115 - exactly 2 = 100
Exactly 2 must = 15

Solution:

We can use the formula:

Total = Group 1 + Group 2 + Group 3 - Exactly 2 Groups - 2 * All 3 Groups + None

Since we need to determine the number of people who play only two sports, we can let “Exactly 2 Groups” = x. Since we are given each person plays at least one sport, “None” = 0. Therefore, we have:

100 = 50 + 45 + 50 - x - 2(15)

100 = 115 - x

x = 15

Answer: C
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Given

• In a class, there are 100 students.
• 50 of them play soccer, 45 basketball, and 50 play volleyball.
• Only 15 play all three games. Everyone plays at least one game.

To Find

• The number of students who play only two games.

Approach and Working Out

• Total count for Soccer is 50, that for basketball is 45 and that for Volleyball is 50.
• People who play one game (A), counted once. People who play two games (B), counted two times and people who play three games (C), counted three times.

o A + B + C = 100
o A + B = 85 (As C = 15) … Equation (1)

• A + 2B + 3C = 50 + 45 + 50 = 145

o A + 2B = 145 – 3 × 15
o A + 2B = 100 … Equation (2)

• From (1) and (2) we get B = 15.

Correct Answer: Option C
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