RTCNTC said:Ten identical apples are to distributed among five of my nephews (A,B,C,D and E). All the ten apples are distributed. How many different ways can the five of my nephews be given apples?
HallsofIvy said:If it is possible that some of the people get no apples, then there are 5 choices who to give the first apple to, 5 choices who to give the second apple to, ... so there are a total of [tex]5^{10}[/tex] choices. If every person must receive an apple, give one apple to each person. Then do there are [tex]5^5[/tex] ways to distribute the other 5 apples.
The answer depends on whether the apples are identical or different. If the apples are identical, then there is only one way to distribute them among the five nephews. However, if the apples are different, then there are 120 different ways to distribute them among the five nephews.
Permutations refer to the different ways in which a set of objects can be arranged. In this case, we can think of the five nephews as the objects and the apples as the different ways they can be arranged. This means that the number of permutations is equal to the number of different ways the apples can be distributed among the five nephews.
Yes, the formula is n!/(n-r)! where n represents the number of objects (in this case, apples) and r represents the number of objects being selected (in this case, nephews). So for five nephews and five apples, the formula would be 5!/(5-5)! = 5!/0! = 5*4*3*2*1 = 120.
In this scenario, combinations and permutations refer to the different ways in which the apples can be distributed among the five nephews. The main difference is that in permutations, the order of the objects matters, while in combinations, the order does not matter. This means that in permutations, giving five apples to nephew A and then four apples to nephew B is considered a different way than giving four apples to nephew B and then five apples to nephew A. In combinations, both of these scenarios would be considered the same way.
One way to visualize this is by using a tree diagram. Start with one apple and then branch out to show the different ways it can be given to the five nephews. Repeat this process for each additional apple until all five apples have been distributed. The total number of branches on the tree will represent the total number of ways the apples can be given to the nephews.