15.) A class has 175 students. The following description gives the number of 100, physics 70, chemistry 46; mathematics and physics 30; mathematics and chemistry 28; physics and chemistry 23; mathematics, physics and chemistry 18. Find (i) how many students are enrolled in mathematics alone, physics alone and chemistry alone, (ii) the number of students who have not offered any of these subjects.
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Answers
Answer:
Ans 60M, 35P, 13C, 22 None.
Here n(U)=175, n(M)=100, n(P)=70
n(C)=46
n(M∩P)=30, n(M∩C)=28
n(P∩C)=28
and n(M∩P∩C)=18
To find n(M∩P
′
∩C
′
), n(P∩M
′
∩C
′
)
n(C∩M
′
∩P
′
)
and n(P
′
∩M
′
∩C
′
)
Now
similarly n(P
′
∩M
′
∩C
′
)=n(P)−n(P∩M)−n(P∩C)+n(P∩M∩C)}
=70−30−23+18=35
n(C∩M
′
∩P
′
)=n(C)−n(C∩M)−n(C∩P)+n(C∩M∩P)}
=46−28−23+18=13
and n(M
′
∩P
′
∩C
′
)=n(M∪P∪C)
′
z
2
ˉ
−
z
1
ˉ
=(
z
2
−z
1
)
=n(U)−{S
1
∩S
2
∩S
3
}
=n(U)−{n(M)+n(P)+n(C)
−[(M∩P)−n(P∩C)
−(M∩C)+n(M∩P∩C)}
=175−100+70+46−30−23−28+18
175−216−81+18=22
Step-by-step explanation:
Ans 60M, 35P, 13C, 22 None.
Here n(U)=175, n(M)=100, n(P)=70
n(C)=46
n(M∩P)=30, n(M∩C)=28
n(P∩C)=28
and n(M∩P∩C)=18
To find n(M∩P
′
∩C
′
), n(P∩M
′
∩C
′
)
n(C∩M
′
∩P
′
)
and n(P
′
∩M
′
∩C
′
)
Now
similarly n(P
′
∩M
′
∩C
′
)=n(P)−n(P∩M)−n(P∩C)+n(P∩M∩C)}
=70−30−23+18=35
n(C∩M
′
∩P
′
)=n(C)−n(C∩M)−n(C∩P)+n(C∩M∩P)}
=46−28−23+18=13
and n(M
′
∩P
′
∩C
′
)=n(M∪P∪C)
′
z
2
ˉ
−
z
1
ˉ
=(
z
2
−z
1
)
=n(U)−{S
1
∩S
2
∩S
3
}
=n(U)−{n(M)+n(P)+n(C)
−[(M∩P)−n(P∩C)
−(M∩C)+n(M∩P∩C)}
=175−100+70+46−30−23−28+18
175−216−81+18=22