You are both correct that there are 5! = 120 permutations of a set of five unique elements {A, B, C, D, E}. Your friend is correct, however, that half of those permutations are the reverse of other permutations in the list. The proof is simple:
Consider any permutation uvxyz. The reverse of this, zyxvu, is also a permutation, and is uniquely determined. Because the reverse is its own inverse, the reverse of zyxvu is uvxyz, the original permutation. Therefore, every permutation in the set of 120 permutations is mirrored by exactly one other permutation, which mirrors no other permutation. We can imagine a bipartite graph whose nodes are labeled with permutations and with arcs connecting nodes whose labels are reverses. The smallest number of nodes which must be removed from this graph to eliminate all edges (mirrorings) is exactly half the nodes in the graph.
To get some intuition about this, imagine first a much smaller case, and then a much larger one. Consider N=3, with set {A, B, C}. There are six permutations: ABC, ACB, BAC, BCA, CAB and CBA. If your friend is right, we'd need to remove three of these; if you were right, we'd remove two. You can verify there are in fact three pairings:
ABC CBA
ACB BCA
BAC CAB
Now, imagine a much larger number, say N=100. If your friend is right, exactly 50% of the permutations would need to be removed. If you were right, only 99!, or exactly 1%, of the permutations would need to be removed. For N=1000, your friend says 50% of permutations must go, and you say only 0.1% are mirrored. And so on.
