Final answer:
To distribute 7 candies to 4 kids where each kid gets at least one candy, we can use the stars-and-bars combinatorial method. After giving one candy to each kid, we distribute the remaining 3 candies, which yields 10 ways to do so. However, this result doesn't match any of the provided answer choices.
Step-by-step explanation:
The question inquires about the number of ways to distribute 7 candies to 4 kids with each kid receiving at least one candy. This is a problem that can be solved using combinations and the stars-and-bars method in combinatorics. First, we give one candy to each kid to fulfill the requirement that every kid must receive at least one candy, leaving us with 7 - 4 = 3 candies to distribute freely. Now, we can think of the problem as placing 3 indistinguishable candies into 4 distinct 'bins' (the kids).
We can use the stars-and-bars method, where the stars represent candies and bars represent separations between kids. For 3 candies (stars) and 3 separations (since there are 4 kids), the formula for combinations with repetition is given by: C(n + k - 1, k), where n is the number of stars, and k is the number of bars.
Thus, for 3 stars and 3 bars, we have: C(3 + 3 - 1, 3) = C(5, 3). The number of combinations is C(5, 3) which is equal to 5! / (3! * (5-3)!) = (5*4*3*2*1) / (3*2*1*2*1) = 10. Therefore, there are 10 ways to distribute the remaining 3 candies. However, none of the answer options provided (a) 35 ways, (b) 21 ways, (c) 15 ways, (d) 9 ways match the calculated number of ways, so there could be a mistake in the options provided.