In a certain examination, $72$ candidates offered Mathematics, $64$ offered English, and $62$ French. $18$ offered both Mathematics and English, $24$ Mathematics and French and $20$ English and French. $8$ candidates offered all the three subjects. How many candidates were there for the examination?

Full solution

Q. In a certain examination,

72

candidates offered Mathematics,

64

offered English, and

62

French.

18

offered both Mathematics and English,

24

Mathematics and French and

20

English and French.

8

candidates offered all the three subjects. How many candidates were there for the examination?

Denote Candidates and Subjects: Let's denote the number of candidates who offered Mathematics as $M$ , English as $E$ , and French as $F$ . According to the problem, we have: $\newline$ $M = 72$ $\newline$ $E = 64$ $\newline$ $F = 62$ $\newline$ The number of candidates who offered both Mathematics and English is $M\cap E = 18$ , both Mathematics and French is $M\cap F = 24$ , and both English and French is $E\cap F = 20$ . The number of candidates who offered all three subjects is $M\cap E\cap F = 8$ . $\newline$ We will use the principle of inclusion-exclusion to find the total number of candidates ( $E$ $0$ ). The formula is: $\newline$ $E$ $1$
Apply Inclusion-Exclusion Principle: Now let's plug in the values we have into the formula: $\newline$ $N = 72 + 64 + 62 - (18 + 24 + 20) + 8$
Calculate Total Number: Perform the calculations inside the parentheses first: $\newline$ $N = 72 + 64 + 62 - 62 + 8$
Calculate Total Number: Perform the calculations inside the parentheses first: $\newline$ $N = 72 + 64 + 62 - 62 + 8$ Now, simplify the expression by adding and subtracting the numbers: $\newline$ $N = 72 + 64 + 8$ $\newline$ $N = 144$