Distributing Four Different Dolls Among Three Children: An Exploration Using Inclusion-Exclusion Principle

In combinatorics, the problem of distributing n different items among k children, where each child must receive at least one item, presents an interesting challenge. This article explores how to solve the specific problem of distributing four different dolls among three children, ensuring that each child gets at least one doll. We'll use the inclusion-exclusion principle to find the number of valid distributions.

1. Introduction to the Problem

The problem is as follows: four different dolls are to be distributed among three children in such a way that each child must receive at least one doll. How many different ways can this be done?

2. Step-by-Step Solution Using Inclusion-Exclusion Principle

Step 1: Total Distributions without Restriction

First, we calculate the total number of ways to distribute the dolls without any restrictions. Since each of the four dolls can be given to any of the three children, the total number of distributions is:

34 81

Step 2: Exclude Cases Where at Least One Child Gets No Doll

Next, we need to exclude the cases where at least one child does not receive a doll. We use the inclusion-exclusion principle to account for this.

2.1 Case 1: One Child Gets No Doll

If we choose one child to receive no doll, the remaining two children can receive the four dolls. The number of ways to choose which child receives no doll is:

( binom{3}{1} )

And for the remaining two children, there are:

24

ways to distribute the dolls. Therefore, for this case:

Ways with one child getting no doll ( binom{3}{1} cdot 2^4 3 cdot 16 48 )

2.2 Case 2: Two Children Get No Dolls

If we choose two children to receive no dolls, then all four dolls must go to the remaining one child. The number of ways to choose which two children get nothing is:

( binom{3}{2} )

And there is only one way to give all dolls to the remaining child. Therefore, for this case:

Ways with two children getting no dolls ( binom{3}{2} cdot 1^4 3 cdot 1 3 )

Step 3: Apply Inclusion-Exclusion Principle

Now we can apply the inclusion-exclusion principle to find the number of valid distributions:

Valid distributions Total distributions - Ways with one child getting no doll - Ways with two children getting no dolls

Substituting the values we calculated:

Valid distributions 81 - 48 - 3 36

3. Alternative Solutions

Alternative Solution 1: Using Combinatorial Distribution

Another approach to solving this problem is to consider that the only way to distribute four dolls among three children with each child getting at least one doll is to have one child receive two dolls and the other two children each receive one doll.

First, choose which child gets two dolls. There are 3 options for which child this is. Then, choose which 2 out of the 4 dolls that this child will receive. There are ( binom{4}{2} ) ways to choose these two dolls. The next child has 2 remaining dolls to choose from, so there are 2 ways to choose the doll for the second child. The last doll is given to the last child.

The total number of ways to distribute the dolls using this method is:

3 × ( binom{4}{2} ) × 2 36

Alternative Solution 2: Inclusion-Exclusion Principle with Events

A third way is to consider the total number of ways to give the dolls, then subtract the cases where at least one child gets no doll. We denote the events as follows:

( A ) - Person 1 gets 0 dolls ( B ) - Person 2 gets 0 dolls ( C ) - Person 3 gets 0 dolls

Using the inclusion-exclusion principle, we calculate:

( P(A cup B cup C) P(A) P(B) P(C) - P(A cap B) - P(A cap C) - P(B cap C) P(A cap B cap C) )

Calculate the individual probabilities:

( P(A) P(B) P(C) 2^4 ) ( P(A cap B) P(A cap C) P(B cap C) 1^4 ) ( P(A cap B cap C) 0 )

Substitute these values into the inclusion-exclusion formula:

( P(A cup B cup C) 3 cdot 2^4 - 3 cdot 1^4 0 48 - 3 45 )

Finally, subtract from the total number of distributions:

34 - 45 81 - 45 36

This method also leads to the same conclusion.

4. Conclusion

Thus, the total number of ways to distribute the four different dolls among three children, with each child receiving at least one doll, is:

36

This problem elegantly demonstrates the power of the inclusion-exclusion principle in combinatorial mathematics and how different approaches can lead to the same solution.

5. Keywords

This article covers the following keywords:

Inclusion-Exclusion Principle Combinatorial Distribution Problem-Solving Techniques