Some problems require finding all permutations (different orderings) of a set of items. For a set of n items { ci,, a2, . . .an } there are n! permutations. For example, given the set {1, 2, 3} there are six permutations:
{3, 2, 1} {2, 3, 1} {2, 1, 3} {3, 1, 2} {1, 3, 2} {1, 2, 3}
Write a recursive function that generates all the permutations of a set of numbers. The general outline of a solution is given here, but the implementation is up to you. The program will require storing a set of permutations of numbers drat you can implement in many ways (for example, linked lists of nodes, linked lists of vectors, arrays, etc.) Your program should call the recursive function with sets of several different sizes, printing the resulting set of permutations for each.
One solution is to first leave out die n th item in the set. Recursively find all permutations using die set of (fl-1) items. If we insert the nth item into each position for all of these permutations, then we get a new set of permutations that includes the n th item. The base case is when there is only one item in the set, in which case the solution is simply the permutation with the single item.
For example, consider finding all permutations of {1, 2, 3}. We leave the 3 out and recursively find all permutations of the set {1, 2}. This consists of the permutations:
{1, 2} {2, 1}
Next we insert the 3 into every position for these permutations. For the first permutation, we insert the 3 in the front, between 1 and 2, and after 2. For the second permutation, we insert the 3 in the front, between 2 and 1, and after 1:
{3, 1, 2} {1, 3, 2} {1, 2, 3} {3, 2, 1} {2, 3, 1} {2, 1, 3}
The resulting six permutations comprise all permutations of the set {1, 2, 3}.
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