एक पंक्ति में 6 व्यक्ति हैं । एक अन्य व्यक्ति को उनमें से 3 व्यक्तियों से इस प्रकार हाथ मिलाना है कि वह दो क्रमागत व्यक्तियों से हाथ नहीं मिलाएगा । ऐसे कितने भिन्न संभाव्य संयोजनों में हाथ मिलाए जा सकते हैं ?

Correct Option: (b) 4

There are 6 persons in a row, let's denote them by positions 1, 2, 3, 4, 5, 6. Another person has to shake hands with 3 of them such that no two consecutive persons are chosen. This is equivalent to choosing 3 non-consecutive items from 6 items. Let the chosen positions be represented by indices from {1, 2, 3, 4, 5, 6}. We need to find the number of combinations of 3 integers out of {1, 2, 3, 4, 5, 6} with no two consecutive. The total number of ways to choose 3 persons out of 6 is 6C3 = 20. Out of these, we subtract the cases where at least two consecutive persons are chosen. Let's list combinations without consecutive elements: {1, 3, 5}, {1, 3, 6}, {1, 4, 6}, {2, 4, 6}. Let's verify each: {1, 3, 5} has difference 2, 2 (valid). {1, 3, 6} has difference 2, 3 (valid). {1, 4, 6} has difference 3, 2 (valid). {2, 4, 6} has difference 2, 2 (valid). Are there any others? Let's check {1, 3, 5}, {1, 3, 6}, {1, 4, 6}, {2, 4, 6}. Total count is 4. Let's use the formula for selecting k non-consecutive items out of n items: C(n - k + 1, k). Here n = 6, k = 3. C(6 - 3 + 1, 3) = C(4, 3) = 4. Thus, there are 4 distinct possible combinations. Therefore, option (b) is correct.