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?

maximum Dec 14, 2023

#1**0 **

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.

BuiIderBoi Dec 17, 2023