Three subjects

In a class of 40 students, 18 passed mathematics, 19 passed accounting, 16 passed economics, five mathematics and accounting only, six mathematics only, nine accounting only, and two accounting and economics only. Each student was offered at least one of the subjects.

a) how many students failed in all the subjects?
b) find the percentage number who failed in at least one of economics and mathematics.
c) calculate the probability that a student selected at random failed in accounts

Correct answer:

a =  4
b =  32.5 %
c =  52.5 %

Step-by-step explanation:

$$\begin{gathered} n=40 \\ M=17 \\ MO=6 \\ A=19 \\ AO=9 \\ E=16 \\ MA=5 \\ EA=2 \\ MAE=A-AO-MA-EA=19-9-5-2=3 \\ ME=M-MO-MA-MAE=17-6-5-3=3 \\ EO=E-ME-MAE-EA=16-3-3-2=8 \\ OK=MO+EO+AO+MAE+MA+EA+ME=6+8+9+3+5+2+3=36 \\ a=n-OK=40-36=4 \end{gathered}$$

$$\begin{gathered} b_2=MO+EO+MA+ME+MAE+EA=6+8+5+3+3+2=27 \\ b=100\cdot \frac{n-b_2}{n}=100\cdot \frac{40-27}{40}=\frac{65}{2}\%=32.5\% \end{gathered}$$

$c=100\cdot \frac{n-A}{n}=100\cdot \frac{40-19}{40}=\frac{105}{2}\%=52.5\%$

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