Six children are each offered a single scoop of any of $3$ flavors of ice cream from the Combinatorial Creamery. In how many ways can each child choose a flavor for their scoop of ice cream so that some flavor of ice cream is selected by exactly five children?

 Dec 14, 2023

To approach this problem, we can break it down into different scenarios:


1. One flavor chosen by five children:


Choose five children out of six to have the same flavor: ⁶C₅ ways.

Choose the flavor for these five children: 3 ways (as there are three options).

The remaining child can choose any of the remaining two flavors: 2 ways.


Total for this scenario: ⁶C₅ * 3 * 2 = 360 ways.


2. Two flavors chosen by two and four children:


There are three ways to choose which pair of flavors will be chosen by two children (flavor A-B, flavor A-C, or flavor B-C).

For each chosen pair, choose two children out of six for that flavor: ⁶C₂ ways.


The remaining four children will take the leftover flavor, meaning two will have that flavor and two will have one of the remaining options: ⁴C₂ ways.


Total for one chosen pair: 3 * ⁶C₂ * ⁴C₂ = 270 ways. Therefore, this scenario contributes 270 * 3 = 810 ways.


3. All three flavors chosen by two children each:


Choose two children each for the three flavors: ⁶C₂ * ⁶C₂ * ⁶C₂ = 6 × 5 × 4 ways.


Total for this scenario: 6 × 5 × 4 = 120 ways.


Total number of ways:


Adding the possibilities for each scenario: 360 + 810 + 120 = 1290 ways.


Therefore, in 1290 ways, six children can choose their ice cream flavors, with one flavor being chosen by exactly five children.

 Dec 17, 2023

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