Men is a count noun, so the 'correct' answer is wrong. Should be 'the number of men'.
I agree completely with Kasper and rhhpk. Apart from the sentence being odd and never heard, the best translation into Danish I think would be 'antallet af mænd'. I can only think og one example where you could say 'mængden af mænd': If you watch a big crowd of people you might say: 'Mængden af mænd er større end mængden af kvinder' (the amount of men is bigger than the amount of women). But then you are talking about 'volume' rather than numbers. Never ever say 'summen af mænd', that's a no-go!