There are 74 students in the Trevel Club. They discovered that 40 members have visited country A, 31 have visited country B, 25 have been to country C, 14 have visited countries A and B, 10 have been only to country C, and 20 have been only, to country A. Some club members have not been to any of the three foreign countries, and, curiously, an equal number have been to all three countries. (a) How many students have been to all three countries? (b) How many students have been only to country B? (e) How many students have been to all three countries?

To solve the problem, let's denote: - A: Students who visited country A - B: Students who visited country B - C: Students who visited country C - x: Students who visited all three countries (A, B, and C) From the information provided, we know: - |A| = 40 (total who visited A) - |B| = 31 (total who visited B) - |C| = 25 (total who visited C) - |A ∩ B| = 14 (visited both A and B) - |C only| = 10 (visited only C) - |A only| = 20 (visited only A) 1. To find out how many students have visited all three countries (x), we can use the inclusion-exclusion principle. From |A|, we can break it down: - Students who visited only A = 20 - Students who visited A and B but not C = |A ∩ B| - x = 14 - x - Students who visited A and C but not B = ? - Students who visited all three (A, B, and C) = x So, for the total in A: \[ |A| = |A \text{ only}| + |A \cap B \text{ only}| + |A \cap C \text{ only}| + |A \cap B \cap C| \] \[ 40 = 20 + (14 - x) + (y) + x \] Let's represent students visiting only A and C as (y). Consequently, we have: \[ 40 = 20 + 14 - x + y + x \] Thus: \[ 40 = 34 + y \] This implies: \[ y = 6 \] Now, breaking it down further for B: - Students only in B = ? - Students who visited B and C but not A = ? The total for B can be expressed as: \[ |B| = |B \text{ only}| + |A ∩ B \text{ only}| + |B ∩ C \text{ only}| + |A \cap B \cap C| \] This gives: \[ 31 = |B \text{ only}| + (14 - x) + (6 - x) + x \] By inserting the known values, we can summarize: \[ 31 = |B \text{ only}| + 14 - x + 6 - x + x \] This leads to: \[ |B \text{ only}| + 20 - x = 31 \] Thus: \[ |B \text{ only}| = 31 - 20 + x = 11 + x \] At this point, the information indicates an equal number of students visited all three countries. Therefore: \[ x = 7 \] So: (a) The number of students who have been to all three countries is **7**. (b) The number of students who have been only to country B is **11** since |B only| = 11 + 7 = 18. (e) For clarity, the number again confirms that the student count for all three countries is **7**.