You have seven balls labelled 1,2,…,7 and seven unlabelled, indistinguishable boxes. How many different ways can the balls be placed into the boxes? Can anyone please work the complete solution. I got the answer but it was solved by my brother and he challenged me to get the complete solution ... Please ...
The boxes are indistinguishable, which means that two configurations are identical if they can be obtained from each other by permuting the boxes.
If you label the boxes by the number of balls they contain, then it is clear that any partition of 7, see here for partitions:
http://en.wikipedia.org/wiki/Partition_(number_theory)
defines a valid configuration and vice versa (because of the permutation symmetry). So, the answer is the number of partitions of 7, which is 15.
To solve this problem, we can use the concept of stars and bars, which is a combinatorial method used to count certain types of arrangements.
In this case, we have seven balls and seven boxes. We want to find the total number of ways we can distribute the balls into the boxes.
Imagine representing the balls with seven stars (*) and representing the boxes with six bars (|). For example, if we have two balls in the first box, one in the second box, and so on, we can represent it as:
**|*||
This representation ensures that there are exactly seven stars and six bars.
Any arrangement of stars and bars corresponds to a unique distribution of balls into boxes. For example, another arrangement could be:
*||||**|
To count the number of possible arrangements, we just need to choose the positions for the six bars among the seven stars. This can be done in C(7+6, 6) ways, where C(n, k) represents the binomial coefficient, which is calculated as:
C(n, k) = n! / (k! * (n-k)!)
Applying the binomial coefficient to our problem, we have:
C(7+6, 6) = C(13, 6) = 13! / (6! * 7!)
Calculating this expression gives us the answer:
13! = 13 * 12 * 11 * 10 * 9 * 8 * 7!
C(13, 6) = 13 * 12 * 11 * 10 * 9 * 8 / (6 * 5 * 4 * 3 * 2 * 1)
C(13, 6) = 1716
Therefore, there are 1716 different ways to place the balls into the boxes.