John's Cupcake Challenge: Finding the Perfect Distribution

In summary, John can give an odd number of cupcakes to each of his 5 students in 17 different ways by dividing 31 cupcakes into 18 stacks of 2 cupcakes each and placing 4 dividers (bars) between them, resulting in 17 spaces. This can be calculated using the combination formula 17c4.
  • #1
Monoxdifly
MHB
284
0
John has baked 31 cupcakes for 5 different students. He wants to give them all to his students but he wants to give an odd number of cupcakes to each one. How many ways can he do this?

Brute-forcing will take about a whole day, I think. If 4 students receive 1 cupcake and the other one receive 27, that's already 4 combinations. If there are 3 1's, the other two might be 3 and 25, 5, and 23, 7 and 21, etc. Is there more efficient way?
 
Mathematics news on Phys.org
  • #2
Hey Mr. Fly,

It's a variation of the stars and bars problem.
See the linked article how it works.

In this particular case we can add 5 cakes first for a total of 36.
Next we divide them in 18 stacks of 2 cakes each.
If we put 4 dividers (bars) in between them, we get 5 portions. We give them to each of the 5 students.
Oh, and before we do so, we take away 1 cake from each portion, so that each student gets an odd number.

How many ways to divide 4 bars over the 17 spaces between the stacks?
 
  • #3
klaas van aarsen said:
how many ways to divide 4 bars over the 17 spaces between the stacks?

17c4?
 
  • #4
Monoxdifly said:
17c4?

Yep. (Nod)
 

FAQ: John's Cupcake Challenge: Finding the Perfect Distribution

1. What is the purpose of John's Cupcake Challenge?

The purpose of John's Cupcake Challenge is to find the most efficient and effective distribution method for his cupcake business. John wants to ensure that his cupcakes are delivered to customers in a timely manner while minimizing costs.

2. Why is finding the perfect distribution important for John's cupcake business?

Finding the perfect distribution is important for John's cupcake business because it can greatly impact his profits and customer satisfaction. The right distribution method can save time and money, while also ensuring that customers receive their cupcakes fresh and in good condition.

3. What factors should John consider when choosing a distribution method?

John should consider factors such as distance to customers, transportation costs, delivery time, and the fragility of his cupcakes. He should also take into account any potential obstacles, such as traffic or weather, that could affect the delivery process.

4. How can John determine the most efficient distribution method?

John can determine the most efficient distribution method by conducting a cost-benefit analysis. This involves comparing the costs and benefits of different distribution methods, such as using a third-party delivery service or delivering the cupcakes himself. He can also gather data and feedback from customers to see which method is most effective.

5. How often should John re-evaluate his distribution method?

John should re-evaluate his distribution method regularly, especially if he notices any issues or changes in his business. It's important to continuously assess and improve the distribution process to ensure maximum efficiency and customer satisfaction.

Similar threads

Replies
1
Views
2K
Replies
15
Views
5K
Replies
9
Views
4K
Replies
125
Views
17K
Replies
1
Views
833
3
Replies
93
Views
12K
Back
Top