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A gathering of $50$ linguists discovered that $4$ knew Kannada$,$ Telugu and Tamil$,$ $7$ knew only Telugu and Tamil $,$ $5$ knew only Kannada and Tamil $,$ $6$ knew only Telugu and Kannada$.$ If the number of linguists who knew Tamil is $24$ and those who knew Kannada is also $24,$ how many linguists knew only Telugu$?$

1. $9$
2. $10$
3. $11$
4. $8$

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Let  $K_a$ denote Kannada, $T_a$ denote Tamil and $T_e$ denote Telegu.

$n(K_a \cup T_a \cup T_e ) = n(K_a) + n(T_a) + n(T_e) - n(K_a \cap T_a ) - n(K_a \cap T_e ) - n(T_e \cap T_a ) + n(K_a \cap T_a \cap T_e )$

$\implies 50 = 24 +24 + n(T_e) - (5+4) - (6+4) - (7+4) + 4$

$\implies 50 = 48 + n(T_e) - 26$

$\implies n(T_e) = 28.$

$\therefore$ Number of people who know Telegu, $n(T_e) =28.$

Number of people who know only Telegu

$= n(T_e) - n(K_a \cap T_e ) - n(T_e \cap T_a ) + n(K_a \cap T_a \cap T_e )$

$= 28 -(6+4) -(7+4) +4$

$= 11$

Correct Option: C.
by Veteran (416k points)

Please correct me if i am wrong.

Let  Ka -> kannada , Ta -> tamil and Te -> telegu.

n(Ka $\cup$ Ta $\cup$ Te ) = n(Ka) + n(Ta) + n(Te) - n(Ka $\cap$ Ta ) - n(Ka $\cap$ Te ) - n(Te $\cap$ Ta ) + n(Ka $\cap$Ta $\cap$ Te )

$\Rightarrow$  50 = 24 +24 + n(Te) - 5 - 6 - 7 + 4

$\Rightarrow$  50 = 48 + n(Te) - 14

$\Rightarrow$ n(Te)  = 2 + 14 =16.

$\therefore$ people who know telegu n(Te) =16

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SO people who know only telegu

= n(Te) - n(Ka $\cap$ Te ) - n(Te $\cap$ Ta ) + n(Ka $\cap$Ta $\cap$ Te )

= 16 -6 -7 +4

= 7

7 is not in the option.

by Boss (17.5k points)