How Do You Calculate the Distribution of 15 Gifts Among People and Parcels?

In summary, for (a) there are 3003 ways to distribute 15 gifts equally amongst Claire, Alana, and Kalena. For (b) there are 3003/6 = 500 ways to distribute 15 gifts into three parcels of five gifts each.
  • #1
ms. confused
91
0
In how many ways can 15 gifts be distributed equally:

a) amongst Claire, Alana, and Kalena

b) into three parcels of five gifts each


For (a) I went [tex]_{15} P_{3}/3 = 910[/tex]

I am 100% certain this is wrong. I also have no idea how to do (b). I would greatly appreciate any help on this question. :cry:
 
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  • #2
Is order important? That is, does it matter which was the first present or is it just a matter of who get what present. If order is important, then it is a permutation problem. If not, then it is a combination problem.
 
  • #3
ms. confused said:
In how many ways can 15 gifts be distributed equally:

a) amongst Claire, Alana, and Kalena

b) into three parcels of five gifts each


For (a) I went [tex]_{15} P_{3}/3 = 910[/tex]

I am 100% certain this is wrong. I also have no idea how to do (b). I would greatly appreciate any help on this question. :cry:
a) from the 15 gifts, first choose 5 from the 15 for Claire, then 5 from the remaining 10 for Alana, and then 5 from the remaining 5 for Kalena. number ways would then be (since order within each choosing of 5 does not matter):
[tex] \mathbb{C}_{5}^{15} \cdot \mathbb{C}_{5}^{10} \cdot \mathbb{C}_{5}^{5} \ = \ (3003) \cdot (252) \cdot (1) [/tex]
b) solution would be similar except order of choosing 1st for Claire, 2nd for Alana, & 3rd from Kalena no longer matters. (of course, the choosing order of the 5 within each group still does not matter). so just divide answer (a) by (3!) to remove the ordering among Claire, Alana, & Kalena to produce 3 parcels of 5 gifts each.
 

FAQ: How Do You Calculate the Distribution of 15 Gifts Among People and Parcels?

What is the difference between permutations and combinations?

Permutations are arrangements of items in a specific order, while combinations are selections of items without regard to order. For example, the permutations of the letters "ABC" would be "ABC," "ACB," "BAC," "BCA," "CAB," and "CBA," while the combinations would be "ABC," "ACB," "BAC," "BCA," "CAB," and "CBA."

How do I calculate the number of possible permutations or combinations?

The formula for permutations is n! / (n-r)!, where n is the total number of items and r is the number of items being selected. The formula for combinations is n! / (r!(n-r)!), where n and r are the same as in permutations. You can also use a calculator or computer program to calculate these values.

Can you give an example of a real-life application of permutations or combinations?

Permutations can be used in situations where order matters, such as arranging a sequence of songs for a playlist. Combinations can be used in situations where order does not matter, such as selecting a group of people for a committee.

How is the fundamental counting principle related to permutations and combinations?

The fundamental counting principle states that if there are m ways to do one task and n ways to do another task, then there are m x n ways to do both tasks. This principle can be applied to permutations and combinations to determine the total number of possible outcomes in a given situation.

Are there any common mistakes to avoid when working with permutations and combinations?

One common mistake is confusing the order of operations. When using the formulas for permutations and combinations, it is important to follow the order of operations and use parentheses to clearly indicate which numbers are being divided and multiplied. Another mistake is forgetting to account for any repeated items in a group, which can change the total number of possible outcomes.

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