A small community consists of 10 women, each of whom has 3 children. If one woman and one of her children are to be chosen as mother and child of the year, how many different choices are possible?

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BlackTom AI · Accepted Answer

Consider the structure of the family setup: there are 10 distinct women, and each woman has exactly 3 children. 
To form a 'mother and child of the year' pair, we must first choose which woman will be the mother, and then choose one of her three children.
For any fixed woman, there are 3 possible children to pair with her, giving 3 possible pairs for that particular mother. 
Since the choice of mother is independent of the choice of child, we multiply the possibilities: 10 women × 3 children per woman = 30 possible mother-child pairs.
Now, evaluate the answer options:
- Option a (40): This would require either more women or more children per woman than given, which contradicts the problem data. Therefore this overcounts relative to the actual setup.
- Option b (30): Matches the computed total number of distinct mother-child pairs under the given constraints, so it aligns with the combinatorial calculation.
- Option c (None of these): Incorrect because there is a valid count (30) among the options, so this is not appropriate.
- Option d (10): This would correspond to choosing only a mother and no child, or alternatively counting only the mothers, which ignores the required 3 possible children per mother.
- Option e (13): An arbitrary number that does not arise from the straightforward product of 10 and 3, hence not correct.
In summary, the correct counting method is multiplication of independent choices (10 mothers × 3 children each), leading to 30 possible pairs, while the other options reflect misunderstandings such as counting only mothers, miscounting children, or proposing inconsistent totals.

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A small community consists of 10 women, each of whom has 3 children. If one woman and one of her children are to be chosen as mother and child of the year, how many different choices are possible?

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