In a class of 160 students, each of them opt at least one language from among English, Hindi and Sanskrit. It is found that 130 students opt English, 120 students Hindi and 110 Sanskrit. If the students opt either <strong>ONLY</strong> one language or <strong>ALL</strong> three languages, then what is the number of students who study all three languages?

Explanation

Let x be the number of students studying all three languages. Since no student studies exactly two, students either study exactly one or all three. The sum of the individual subject totals is 130 + 120 + 110 = 360, which counts the students studying one language once and the students studying all three languages thrice. Thus, (E+H+S) + 3x = 360. We also know the total is (E+H+S) + x = 160. Subtracting the two gives 2x = 200 \implies x = 100.

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