Eight married couples, that is 8 husbands and their 8 wives, attend a dance show called Nach Baliye. In a lucky draw, four persons are selected at random for a prize. What is the probability that at least one complete married couple is included among the four selected persons?

Introduction / Context:
This problem deals with selecting a group from married couples and asks for the probability that the chosen group of four contains at least one complete couple. It is often easier to solve such problems using the complement approach, first calculating the probability that no couple appears together and then subtracting this from 1.

Given Data / Assumptions:

• There are 8 married couples, so total persons = 16.
• Four persons are selected at random without restriction.
• Each group of four persons is equally likely.
• We want at least one couple in the group, meaning one or more husband wife pairs.

Concept / Approach:
Let Event A be the event that at least one couple is selected. Instead of counting all groups that have one or more couples directly, we compute the probability of the complement event, B: no couple is selected. Then P(A) = 1 minus P(B). To enforce no couple, we ensure that at most one person from each couple is selected.

Step-by-Step Solution:
Total number of 4 person groups from 16 people = C(16, 4) = 1820. To count groups with no couple, first choose which 4 couples contribute one member: C(8, 4). C(8, 4) = 70. From each chosen couple, we can choose either the husband or the wife, so 2 choices per couple. Number of groups with no couple = 70 * 2^4 = 70 * 16 = 1120. Therefore, P(no couple) = 1120 / 1820. P(at least one couple) = 1 - 1120 / 1820 = (1820 - 1120) / 1820 = 700 / 1820. Simplify 700 / 1820: divide numerator and denominator by 140 to get 5 / 13, which equals 15 / 39.

Verification / Alternative check:
Instead of simplifying directly to 5/13, we notice that option 15/39 reduces to 5/13. The other given fractions, when simplified, do not match 5/13. This consistency check confirms that 15/39 represents the correct probability in the given options.

Why Other Options Are Wrong:
8/39 is smaller and does not come from any natural count related to 700 favourable groups. 12/13 is very large and would mean almost every group contains a couple, which is not true. None of these is incorrect because there is a matching fraction 15/39 that equals 5/13.

Common Pitfalls:
A frequent mistake is to try to count groups with at least one couple directly, which involves overlapping cases with one, two or more couples and is complicated. Another error is to miscount the no couple cases by forgetting the factor 2^4 for selecting one person from each chosen couple. Using the complement method with clear combinatorial reasoning keeps the calculation manageable.

Final Answer:
Thus, the probability that at least one married couple is included among the four selected persons is 15/39, which simplifies to 5/13.