Answer to Question #320001 in Discrete Mathematics for Sheriff

Question

There are 60 science students tn a

Secondary School. 35 of whom study

Chemistry and 30 study Technical

Drawing, 12 out of those students atudy

Biology and chemistry but not Technical

Drawing, 10 study Chemistry but

neither biology nor Technical Drawing 11

study only Technical Drawing and

neither Biology nor Chemistry, 10 also

study Chemistry and Technical Drawing

only

a. How many mdents study all the three

subjects?

b. How many students study biology and

Technical Drawing but not Chemistry?

C. How many students study Biology only?

d How many students study biology all

together.

Answer 1

$\varOmega -all\,\,students\\C-chemistry\\T-technical\,\,Drawing\\B-bio\log y\\\left| \varOmega \right|=60,\left| C \right|=35,\left| T \right|=30,\left| C\cap \bar{T}\cap B \right|=12,\left| C\cap \bar{T}\cap \bar{B} \right|=10,\left| \bar{C}\cap T\cap \bar{B} \right|=11,\left| C\cap T\cap \bar{B} \right|=10\\a:\\\left| C\cap T\cap B \right|=\left| C \right|-\left| C\cap \bar{T}\cap \bar{B} \right|-\left| C\cap \bar{T}\cap B \right|-\left| C\cap T\cap \bar{B} \right|=35-10-12-10=3\\b:\\\left| \bar{C}\cap T\cap B \right|=\left| T \right|-\left| C\cap T\cap \bar{B} \right|-\left| C\cap T\cap B \right|-\left| \bar{C}\cap T\cap \bar{B} \right|=30-10-3-11=6\\c:\\\left| \bar{C}\cap \bar{T}\cap B \right|=\left| \varOmega \right|-\left| C\cap T\cap B \right|-\left| C\cap \bar{T}\cap B \right|-\left| \bar{C}\cap T\cap B \right|-\left| C\cap \bar{T}\cap \bar{B} \right|-\left| C\cap T\cap \bar{B} \right|-\left| \bar{C}\cap T\cap \bar{B} \right|=\\=60-3-12-6-10-10-11=8\\d:\\\left| B \right|=\left| \bar{C}\cap \bar{T}\cap B \right|+\left| \bar{C}\cap T\cap B \right|+\left| C\cap \bar{T}\cap B \right|+\left| C\cap T\cap B \right|=8+6+12+3=29$