Step-by-step Solution:

In a room containing 28 people, there are 18 people who speak English, 15 people who speak Hindi, and 22 people who speak Kannada. We are given the following:\[\] - Total number of people = 28 \[\] - People who speak English = 18 \[\] - People who speak Hindi = 15 \[\] - People who speak Kannada = 22 \[\] - People who speak both English and Hindi = 9 \[\] - People who speak both Hindi and Kannada = 11 \[\] - People who speak both Kannada and English = 13 \[\] We are to find how many people speak all three languages.\[\] \[\] Let’s define:\[\] ( E ): Set of people speaking English \[\] - ( H ): Set of people speaking Hindi \[\] - ( K ): Set of people speaking Kannada \[\] - ( E \(\cap\) H \(\cap\) K ): People who speak all three languages \[\] - Given intersections: \[\] ( |E \(\cap\) H| = 9 ) \[\] ( |H \(\cap\) K| = 11 ) \[\] ( |K \(\cap\) E| = 13 ) \[\] We aim to calculate ( |E \(\cap\) H \(\cap\) K| ).\[\] \[\] Using the Principle of Inclusion-Exclusion: The general formula for three sets is given by:\[\] |E \(\cup\) H \(\cup\) K| = |E| + |H| + |K| - |E \(\cap\) H| - |H \(\cap\) K| - |K \(\cap\) E| + |E \(\cap\) H \(\cap\) K| \[\] We are given that the total number of people is 28. Thus:\[\] |E \(\cup\) H \(\cup\) K| = 28 \[\] Substitute the known values:\[\] 28 = |E| + |H| + |K| - |E \(\cap\) H| - |H \(\cap\) K| - |K \(\cap\) E| + |E \(\cap\) H \(\cap\) K| \[\] Substitute the given numbers:\[\] 28 = 18 + 15 + 22 - 9 - 11 - 13 + |E \(\cap\) H \(\cap\) K| \[\] Calculate the constants step-by-step:\[\] 1. Sum of individual sets: ( 18 + 15 + 22 = 55 ) \[\] 2. Subtract the pairwise intersections: ( 9 + 11 + 13 = 33 ) \[\] Now compute:\[\] 28 = 55 - 33 + |E \(\cap\) H \(\cap\) K| \[\] 28 = 22 + |E \(\cap\) H \(\cap\) K| \[\] Solve for ( |E \(\cap\) H \(\cap\) K| ):\[\] |E \(\cap\) H \(\cap\) K| = 28 - 22 \[\] |E \(\cap\) H \(\cap\) K| = 6 \[\] \[\] Final Answer:\[\] The number of people who speak all three languages is:\[\] \[\] Correct Option: (a)