In a class of 32 students, 20 passed in mathematics, 10 passed in phys

1.	In a class of 32 students, 20 passed in mathematics, 10 passed in physics, 15 passed in chemistry, 4 passed in both mathematics and physics, 6 passed in physics and chemistry and 2 students passed in all the three subjects. how many of them passed in both mathematics and chemistry?Pls tell how to solve
Answer» Let M, P and C be the SETS of students taking examinations in MATHEMATICS, Physics and Chemistry, respectively. We have, n(M∪P∪C)=50,n(M)=37,n(P)=24, n(C)=43 n(M∩P)≤19,n(M∩C)≤29,n(P∩C)≤20 Now, n(M∪P∪C)=n(M)+n(P)+n(C)−n(M∩P)−n(M∩C)−n(P∩C)+n(M∩P∩C) ⇒50=37+24+43−{n(M∩P)+n(M∩C)+n(P∩C)}+n(M∩P∩C) ⇒n(M∩P)+n(M∩C)+n(P∩C)=n(M∩P∩C)+54..(i) Now, n(M∩P)+n(M∩C)+n(P∩C)≤19+29+20 [using Eq. (i)] ⇒n(M∩P∩C)+54≤68 ⇒n(M∩P∩C)≤14. Helpful to you

In a class of 32 students, 20 passed in mathematics, 10 passed in physics, 15 passed in chemistry, 4 passed in both mathematics and physics, 6 passed in physics and chemistry and 2 students passed in all the three subjects. how many of them passed in both mathematics and chemistry?Pls tell how to solve

Answer»

Let M, P and C be the SETS of students taking examinations in MATHEMATICS, Physics and Chemistry, respectively.

We have,

n(M∪P∪C)=50,n(M)=37,n(P)=24,

n(C)=43

n(M∩P)≤19,n(M∩C)≤29,n(P∩C)≤20

Now, n(M∪P∪C)=n(M)+n(P)+n(C)−n(M∩P)−n(M∩C)−n(P∩C)+n(M∩P∩C)

⇒50=37+24+43−{n(M∩P)+n(M∩C)+n(P∩C)}+n(M∩P∩C)

⇒n(M∩P)+n(M∩C)+n(P∩C)=n(M∩P∩C)+54..(i)

Now, n(M∩P)+n(M∩C)+n(P∩C)≤19+29+20 [using Eq. (i)]

⇒n(M∩P∩C)+54≤68

⇒n(M∩P∩C)≤14.