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# Binomial help needed

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Please Help! I've tried 6 choose 3 and 6 choose 2 and they are wrong. These are the wrong answers I've treid: 18, 20 15, 120, 363.

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 three children?

Sep 24, 2023

#1
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To find the number of ways in which each child can choose a flavor of ice cream such that exactly three children choose the same flavor, we can break down the problem into cases.

Case 1: Three children choose the same flavor, and the other three children choose different flavors. In this case, we need to select one flavor for the three children who choose the same flavor. There are 3 ways to do that. Then, we need to select three different flavors for the remaining three children, which can be done in 3! = 6 ways. Therefore, the total number of ways for this case is 3 * 6 = 18.

Case 2: Four children choose one flavor, and the other two children choose a different flavor. In this case, we need to select one flavor for the four children who choose the same flavor, which can be done in 3 ways. Then, we need to select one flavor for the two remaining children, which can be done in 2 ways. Therefore, the total number of ways for this case is 3 * 2 = 6.

Case 3: Five children choose one flavor, and the remaining child chooses a different flavor. In this case, we need to select one flavor for the five children who choose the same flavor, which can be done in 3 ways. Then, we need to select one flavor for the remaining child, which can be done in 2 ways. Therefore, the total number of ways for this case is 3 * 2 = 6.

Case 4: All six children choose the same flavor. In this case, we need to select one flavor for all six children, which can be done in 3 ways. Therefore, the total number of ways for this case is 3.

Finally, we sum up the results from all cases to get the total number of ways: 18 + 6 + 6 + 3 = 33.

Therefore, there are 33 ways in which each child can choose a flavor of ice cream so that exactly three children choose the same flavor.

Sep 24, 2023
edited by ianswerquestions  Sep 24, 2023
edited by ianswerquestions  Sep 24, 2023
#2
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That's wrong, sorry!

Keihaku  Sep 24, 2023
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We can approach this problem using the principle of inclusion-exclusion.

First, let's calculate the total number of ways that the children can choose their ice cream flavors without any restrictions. Each child has 3 choices, so there are 3^6 = 729 possible ways.

Next, let's count the number of ways that no flavor is selected by exactly three children. There are three cases to consider:

No flavor is selected by any of the children. In this case, each child has 3 choices, so there are 3^6 = 729 ways.

One flavor is selected by exactly two children, and the other two flavors are selected by two children each. There are three ways to choose the flavor that is selected by two children. Once that flavor is chosen, there are (6 choose 2) ways to choose the two children who will select that flavor, and then the remaining four children each have two choices for their ice cream flavor. So the total number of ways is 3 * (6 choose 2) * 2^4 = 540.

Each flavor is selected by either one or two children. In this case, there are (3 choose 2) ways to choose the two flavors that are selected by two children, and (6 choose 2) ways to choose the two children who will select each of those flavors. The remaining two children each have two choices for their ice cream flavor. So the total number of ways is (3 choose 2) * (6 choose 2) * 2^2 = 540.

Using the principle of inclusion-exclusion, we can now count the total number of ways that some flavor is selected by exactly three children:

Total number of ways = 3^6 - (number of ways no flavor is selected by exactly three children)

= 729 - (729 + 540 + 540 - (number of ways some flavor is selected by exactly two children))

Now we need to count the number of ways that some flavor is selected by exactly two children. There are three ways to choose the flavor that is selected by two children, and (6 choose 2) ways to choose the two children who will select that flavor. The remaining four children each have two choices for their ice cream flavor. So the total number of ways is 3 * (6 choose 2) * 2^4 = 540.

Substituting this value, we get:

Total number of ways = 729 - (729 + 540 - 380) = 160

Sep 25, 2023