Five dogs (Fido, Ruff, Lassie, Odie, and Snoopy) were playing in the park. Altogether, they had 5 identical Frisbees. In how many ways could the 5 Frisbees be distributed among the dogs when they leave the park to go home?
what are the partitions of 5?
0,0,0,0,5
0,0,0,1,4
...
1,1,1,1,1
To solve this problem, we can use the concept of distributing identical objects to distinct groups. Since all 5 Frisbees are identical, we can treat this problem as distributing 5 identical objects among 5 distinct dogs.
To find the number of ways to distribute the Frisbees, we can use the stars and bars method. In this method, we imagine placing the 5 Frisbees in a row and inserting 4 bars between them. Each bar separates the Frisbees and represents a different dog receiving the Frisbees.
For example, one possible distribution could look like this:
Frisbee 1 | Frisbee 2 | Frisbee 3 | Frisbee 4 | Frisbee 5
| | | |
To calculate the number of ways, we need to determine the number of ways we can arrange the 5 Frisbees and 4 bars. In this case, there will be a total of 9 positions (5 Frisbees + 4 bars). We need to choose 4 positions for the bars, and the remaining positions will be occupied by the Frisbees.
Using combinations, the number of ways to distribute the Frisbees is given by:
C(n + r - 1, r) = C(9, 4) = 126
So, there are 126 different ways to distribute the 5 identical Frisbees among the 5 dogs when they leave the park to go home.
I don't get it.
Since all the frisbees are the same, and assuming no dog gets more than 1 frisbee, there would be only 1 way.
If they were different, there would be 5! or 120 ways