Distributing Candies Among Students Using Combinatorial Methods

This article explains how to use combinatorial methods to solve two problems involving the distribution of candies among students. We explore the challenge of distributing all candies to students without restrictions and the specialized case where each student receives exactly two candies. Let's dive into the details of each part.

Introduction to Combinatorial Methods

Combinatorial methods are a powerful tool in solving problems related to counting and distributing items. In this article, we will use combinatorial methods to solve two problems: (a) distributing all 8 different candies to 4 students, and (b) distributing the candies such that each student receives exactly 2 candies.

Part A: Distributing All Candies to Students

In the first part of the problem, the teacher wants to distribute all 8 different candies to 4 students without any restrictions. Each candy can be given to any one of the 4 students.

Combinatorial Calculation

For each candy, there are 4 choices (one for each student). Since there are 8 candies and each candy can be given independently, the total number of ways to distribute the candies is:

( 4^8 )

Calculating this:

( 4^8 65536 )

Therefore, the number of different ways to give all the candies to the students is 65,536.

Part B: Distributing Candies Such That Each Student Gets 2 Candies

In the second part of the problem, the teacher wants to distribute the 8 different candies such that each of the 4 students receives exactly 2 candies. This is a more complex combinatorial problem that involves selecting groups of candies.

Combinatorial Calculation

To solve this problem, we need to calculate the number of ways to choose 2 candies for each student. This can be done using binomial coefficients.

Step-by-Step Calculation

Choose 2 candies for the first student: ( binom{8}{2} )

Choose 2 candies for the second student: ( binom{6}{2} )

Choose 2 candies for the third student: ( binom{4}{2} )

The last 2 candies go to the fourth student: ( 1 )

Thus, the total combinations can be calculated as follows:

( binom{8}{2} times binom{6}{2} times binom{4}{2} times 1 )

Calculating each binomial coefficient:

( binom{8}{2} frac{8 times 7}{2 times 1} 28 )

( binom{6}{2} frac{6 times 5}{2 times 1} 15 )

( binom{4}{2} frac{4 times 3}{2 times 1} 6 )

Now multiplying these together:

( 28 times 15 times 6 2520 )

Since the order of students matters, the same set of candies given to different students counts as a different distribution. We must multiply by the number of ways to arrange the 4 students, which is ( 4! ).

( 4! 4 times 3 times 2 times 1 24 )

Thus, the total number of ways to distribute the candies is:

( 2520 times 24 60480 )

Therefore, the number of different ways to give all the candies such that every student gets 2 candies is 60,480.

Conclusion

In conclusion, this article has detailed the use of combinatorial methods to solve two candy distribution problems. The first problem involved distributing all candies to students without restrictions, yielding 65,536 different ways. The second problem required that each student received exactly two candies, resulting in 60,480 different ways. These methods showcase the power of combinatorial techniques in solving real-world problems.

Further Reading

For further reading on combinatorial methods and their applications, refer to texts on discrete mathematics and probability.