Wa Gritan eximimation, $72$ cedidales opended matheratios, oq apiered Anglish and $C_{2}$ frencle $18$ aploed both mathematios - d Inglsh, $24$ mothemater and Fench ad $20$ ongl'sh and princh $8$ Godidates proed all the theee sublects tow mary Cradidries Were there For the examintion?

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Q. Wa Gritan eximimation,

72

cedidales opended matheratios, oq apiered Anglish and

C_{2}

frencle

18

aploed both mathematios - d Inglsh,

24

mothemater and Fench ad

20

ongl'sh and princh

8

Godidates proed all the theee sublects tow mary Cradidries Were there For the examintion?

Define Sets: Let's define the sets: $\newline$ $M =$ candidates who appeared for Mathematics $\newline$ $E =$ candidates who appeared for English $\newline$ $F =$ candidates who appeared for French $\newline$ According to the problem, we have the following information: $\newline$ $|M| = 72$ (candidates appeared for Mathematics) $\newline$ $|E| =$ number of candidates appeared for English (not directly given) $\newline$ $|F| =$ number of candidates appeared for French (not directly given) $\newline$ $|M \cap E| = 18$ (candidates appeared for both Mathematics and English) $\newline$ $|M \cap F| = 24$ (candidates appeared for both Mathematics and French) $\newline$ $|E \cap F| = 20$ (candidates appeared for both English and French) $\newline$ $|M \cap E \cap F| = 8$ (candidates appeared for all three subjects) $\newline$ We need to find the total number of candidates, which is $E =$ $0$ . $\newline$ We can use the principle of inclusion-exclusion to find this number: $\newline$ $E =$ $1$ $\newline$ However, we do not have the individual numbers for $E =$ $2$ and $E =$ $3$ , so we cannot directly apply the formula. We need to find a way to express $E =$ $2$ and $E =$ $3$ in terms of the given information.
Apply Principle of Inclusion-Exclusion: Let's denote the total number of candidates as $T$ . We know that every candidate appeared for at least one subject. Therefore, we can write: $\newline$ $T = |M| + |E| + |F| - |M \cap E| - |M \cap F| - |E \cap F| + |M \cap E \cap F|$ $\newline$ We can plug in the values we know: $\newline$ $T = 72 + |E| + |F| - 18 - 24 - 20 + 8$ $\newline$ Now we need to simplify the equation: $\newline$ $T = 72 + |E| + |F| - 54 + 8$ $\newline$ $T = 26 + |E| + |F|$ $\newline$ We still need to find $|E|$ and $|F|$ , but we can't do that with the information given. It seems there might be an error in the problem statement or missing information, as we cannot determine the number of candidates who appeared only for English or only for French.