In a group of persons travelling in a bus, 6 persons can speak Tamil, 15 can speak Hindi and 6 can speak Gujarati. In that group none can speak any other language. If 2 persons in the group can speak two languages only and one person can speak all the three languages, then how many persons are there in the group?

The problem can be solved using the principle of inclusion-exclusion for three sets, or by visualizing the distinct regions in a Venn diagram.

Let N(T), N(H), N(G) be the number of people speaking Tamil, Hindi, and Gujarati, respectively. N(T) = 6 N(H) = 15 N(G) = 6

Let N(exactly two languages) be the number of people who can speak exactly two languages. N(exactly two languages) = 2

Let N(all three languages) be the number of people who can speak all three languages. N(all three languages) = 1

The total number of persons in the group (N_total) can be calculated using the formula: N_total = N(T) + N(H) + N(G) - N(exactly two languages) - 2 * N(all three languages)

This formula accounts for the overlaps correctly:

• The sum N(T) + N(H) + N(G) counts people speaking one language once, people speaking two languages twice, and people speaking three languages thrice.
• Subtracting N(exactly two languages) removes one count for each person speaking exactly two languages, so they are now counted once.
• Subtracting 2 * N(all three languages) removes two counts for each person speaking all three languages. Since they were counted thrice initially, after this subtraction, they are counted once.

Substitute the given values into the formula: N_total = 6 + 15 + 6 - 2 - (2 * 1) N_total = 27 - 2 - 2 N_total = 23

Thus, there are 23 persons in the group.

Analyzing the options: A) 21: Incorrect. This would result if an incorrect subtraction was made (e.g., 27 - 6). B) 22: Incorrect. This would result if an incorrect subtraction was made (e.g., 27 - 5). C) 23: Correct. This matches our calculation. D) 24: Incorrect. This would result if an incorrect subtraction was made (e.g., 27 - 3).

The final answer is C.