Iiva Certain eximintion, tz candidales opzexed mathentics, $64$ offered finglish and $G_{2}$ French. $18$ offered both mathemates and Inghsh, $24$ mathematers and French and $20$ - English and prench. $8$ Gudidates eqpered all the three sublects. How many Cundidates Where there For the ceamintion?

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Q. Iiva Certain eximintion, tz candidales opzexed mathentics,

64

offered finglish and

G_{2}

French.

18

offered both mathemates and Inghsh,

24

mathematers and French and

20

- English and prench.

8

Gudidates eqpered all the three sublects. How many Cundidates Where there For the ceamintion?

Denote Candidates: Let's denote the number of candidates who offered mathematics as $M$ , English as $E$ , and French as $G$ . According to the problem, we have the following information: $\newline$ - $M$ candidates offered mathematics. $\newline$ - $64$ candidates offered English. $\newline$ - $G_2$ candidates offered French. $\newline$ - $18$ candidates offered both mathematics and English. $\newline$ - $24$ candidates offered both mathematics and French. $\newline$ - $20$ candidates offered both English and French. $\newline$ - $8$ candidates offered all three subjects. $\newline$ We will use the principle of inclusion-exclusion to find the total number of candidates ( $E$ $0$ ). The formula is: $\newline$ $E$ $1$ $\newline$ Where: $\newline$ - $M$ = candidates who offered mathematics $\newline$ - $E$ = candidates who offered English $\newline$ - $G$ = candidates who offered French $\newline$ - $E$ $5$ = candidates who offered both mathematics and English $\newline$ - $E$ $6$ = candidates who offered both mathematics and French $\newline$ - $E$ $7$ = candidates who offered both English and French $\newline$ - $E$ $8$ = candidates who offered all three subjects $\newline$ We know $E$ $9$ and $G$ $0$ . However, we do not have the values for $M$ and $G$ . We need to find these values using the given information.
Find Candidates Offering Subjects: First, let's find the total number of candidates who offered both mathematics and another subject. We have: $\newline$ $ME + MF - MEF = 18 + 24 - 8$ $\newline$ $ME + MF - MEF = 42 - 8$ $\newline$ $ME + MF - MEF = 34$ $\newline$ This means that $34$ candidates offered mathematics and at least one other subject, excluding those who offered all three.
Express Total Candidates: Next, let's find the total number of candidates who offered both English and another subject. We have: $\newline$ $ME + EF - MEF = 18 + 20 - 8$ $\newline$ $ME + EF - MEF = 38 - 8$ $\newline$ $ME + EF - MEF = 30$ $\newline$ This means that $30$ candidates offered English and at least one other subject, excluding those who offered all three.
Find Values for M and G_2: Now, let's find the total number of candidates who offered both French and another subject. We have: $\newline$ $MF + EF - MEF = 24 + 20 - 8$ $\newline$ $MF + EF - MEF = 44 - 8$ $\newline$ $MF + EF - MEF = 36$ $\newline$ This means that $36$ candidates offered French and at least one other subject, excluding those who offered all three.
Problem Statement Missing: We can now express the total number of candidates $T$ as: $\newline$ $T = M + 64 + G_2 - (34 + 30 + 36) + 8$ $\newline$ $T = M + 64 + G_2 - 100 + 8$ $\newline$ $T = M + G_2 - 28$ $\newline$ We still need to find the values for $M$ and $G_2$ to solve for $T$ .
Problem Statement Missing: We can now express the total number of candidates $T$ as: $\newline$ $T = M + 64 + G_2 - (34 + 30 + 36) + 8$ $\newline$ $T = M + 64 + G_2 - 100 + 8$ $\newline$ $T = M + G_2 - 28$ $\newline$ We still need to find the values for $M$ and $G_2$ to solve for $T$ .The problem does not provide explicit values for $M$ and $G_2$ , and without these values, we cannot determine the total number of candidates $T$ . It seems there is a mistake or missing information in the problem statement, as we cannot solve for $T$ with the given information.