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

[ms. confused]

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 $$_{15} P_{3}/3 = 910$$

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.

 

[HallsofIvy]

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.

 

[geosonel]

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 $$_{15} P_{3}/3 = 910$$

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.

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):
$$\mathbb{C}_{5}^{15} \cdot \mathbb{C}_{5}^{10} \cdot \mathbb{C}_{5}^{5} \ = \ (3003) \cdot (252) \cdot (1)$$
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.