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IN a Cartain exmimation, 72 candidates ofzered methenatics, 64 affered finglish and
G_(2) French, 18 afianed 5oth makremales ind Inglish, 24 mathematies and French and 20 -Cuglish and erench. 8 Gndidates eppered rll the three sublects. How mary Cindidates where there For the examintion?

IN a Cartain exmimation, $72$ candidates ofzered methenatics, $64$ affered finglish and $G_{2}$ French, $18$ afianed $5$ oth makremales ind Inglish, $24$ mathematies and French and $20$ -Cuglish and erench. $8$ Gndidates eppered rll the three sublects. How mary Cindidates where there For the examintion?

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GCF and LCM: word problems

Full solution

Q. IN a Cartain exmimation,

72

candidates ofzered methenatics,

64

affered finglish and

G_{2}

French,

18

afianed

5

oth makremales ind Inglish,

24

mathematies and French and

20

-Cuglish and erench.

8

Gndidates eppered rll the three sublects. How mary Cindidates where there For the examintion?

Identify sets and relationships: Identify the sets and their relationships. $\newline$ We have three sets: candidates who offered Mathematics $M$ , English $E$ , and French $F$ . Some candidates offered more than one subject, and some offered all three. We need to use the principle of inclusion-exclusion to find the total number of candidates.
Write given numbers: Write down the given numbers. $\newline$ Number of candidates who offered Mathematics (M) = $72$ $\newline$ Number of candidates who offered English (E) = $64$ $\newline$ Number of candidates who offered French (F) = $G_{2}$ $\newline$ Number of candidates who offered both Mathematics and English $(M \cap E)$ = $18$ $\newline$ Number of candidates who offered both Mathematics and French $(M \cap F)$ = $24$ $\newline$ Number of candidates who offered both English and French $(E \cap F)$ = $20$ $\newline$ Number of candidates who offered all three subjects $(M \cap E \cap F)$ = $64$ $0$
Apply inclusion-exclusion principle: Apply the principle of inclusion-exclusion. $\newline$ Total number of candidates = $|M| + |E| + |F| - |M \cap E| - |M \cap F| - |E \cap F| + |M \cap E \cap F|$
Substitute values into formula: Substitute the given values into the formula. $\newline$ Total number of candidates = $72 + 64 + G_{2} - 18 - 24 - 20 + 8$
Simplify the expression: Simplify the expression. $\newline$ Total number of candidates = $72 + 64 + G_{2} - 18 - 24 - 20 + 8$ $\newline$ Total number of candidates = $136 + G_{2} - 54$ $\newline$ Total number of candidates = $82 + G_{2}$
Identify value of $G_{2}$ : Identify the value of $G_{2}$ . $\newline$ The problem statement does not provide a numerical value for $G_{2}$ , which represents the number of candidates who offered French. Without this value, we cannot calculate the exact total number of candidates.
Conclude with given information: Conclude with the information given. $\newline$ Since $G_{2}$ is not provided, we can only express the total number of candidates as a function of $G_{2}$ . The total number of candidates for the examination is $82$ plus the number of candidates who offered French.

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