Number of children visiting an ice cream shop when 9 flavours are available and each child takes a cone with a unique pair of flavours

Introduction / Context:
This question uses combinations to model how many distinct two flavour cones can be made from a set of flavours. Each child chooses a cone with two distinct flavours, and no two children choose the same pair, so the number of children equals the number of unique two flavour combinations.

Given Data / Assumptions:

• There are 9 distinct ice cream flavours available.
• Each child chooses exactly one cone.
• Each cone contains exactly two different flavours.
• No two children choose the same combination of flavours.
• Order of flavours on the cone does not matter, so flavour A with flavour B is the same as flavour B with flavour A.

Concept / Approach:
Each cone is identified by a combination of 2 distinct flavours chosen from 9. Since order does not matter, the correct count is given by a combination, not a permutation. The number of children is equal to the number of such distinct combinations.

Step-by-Step Solution:
Step 1: Let the number of flavours be n = 9.Step 2: Each cone picks 2 distinct flavours, so the number of possibilities is 9C2.Step 3: Use the combination formula nC2 = n * (n - 1) / 2.Step 4: Substitute n = 9 to get 9C2 = 9 * 8 / 2.Step 5: Compute 9 * 8 = 72 and divide by 2 to get 36.Step 6: No two children can have the same combination, so the maximum number of children is 36.

Verification / Alternative check:
You can list pairs in groups to verify the count. For example, pair the first flavour with the remaining 8, the second with the next 7, and so on. The total pairs become 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 36, which matches 9C2.

Why Other Options Are Wrong:

• 28, 44, and 56 are other combination values from different nC2 cases but do not match 9C2.
• 56 would be the number of pairs from 8 distinct flavours plus some incorrect extra counting.

Common Pitfalls:
A common error is to use permutations 9P2 = 72, which would treat flavour order as important and overcount each cone by a factor of 2. Another mistake is to misread the question and include combinations with repetition, such as two scoops of the same flavour, which is not allowed here.

Final Answer:
The number of children who can each have a unique two flavour cone is 36, so the correct answer is 36.