Ideally you should never encounter a situation where you'd have to "ask" for a dance. It is best to try any of the alternatives provided below before resorting to the demeaning and supplicating "Would you like to dance with me?"
You can use Mr Smooth to start a dance with a group of girls. Or be on the dance-floor by yourself manoeuvring into position with a girl / a group of girls dancing by themselves. Or you can dance near the edge of the dance-floor and pull any girls passing by to dance with you. Or you can act like you already know the girl, take her hand, nod towards the dancfloor and say "well, come on!:)". If she seems reluctant, you go "oh, come on!:)". If she says "I'm tired", BOOM, you're in a conversation (you didn't even notice that, did you?:). "Tired? Well, not that you mention it, this place is a little crowded/noisy. Why don't we go to someplace quieter where we can talk and you can relax:)" If you let go of her hand in the meantime, grab it again and guide her to a nice and quiet little corner you've already checked out before:)
Or - you can go the classic AFC way of asking a girl for a dance. If she says yes, great! If however she refuses, you're out of luck. You can try and salvage the situation by joining her for a conversation but she has refused you once already so a come-back will be that much harder to do. Or you can just move on, but remember - having other girls see you get rejected diminishes your chances with them even further. Which could result in the ultimate humiliation of moving from girl to girl and asking for a dance and being turned down all the way until you run out of girls:) Ok, the latter is unlikely, someone WILL eventually dance with you, but most probably that someone will be (yep, you guessed it:) - an UG!:)
Ideally, rejection should never happen, for if you've had a chance to talk to her in a meaningful manner before, she is already dying to have a little kino and grinding with you on the dance-floor:) But amongst the deafening and blinding circumstances of a typical night-club, this can be hard to do. So she just takes you by face-value, and its gonna be a yes or no.
Also, when doing an all-out "asking for a dance", some annoying factors might come to play, over which you have very little control owing to the approach you have chosen. She might be tired, waiting for a friend in a designated spot, just doesn't dance, her boyfriend is just around the corner. All these factors really have no bearing should you choose an approach which also incorporates demonstrating value and personality (beyond having the guts to ask for a dance). In which case "waiting for a friend" or "being tired" are quickly out the window... plus some more time with you and the "boyfriend" might share the same route:)
But without having been able build rapport with her and make her crave for you, you really don't have too many options - maybe thank her, maybe ask her "how about later?" (yea right) or ask her "mind if I join you for some conversation?", nah, these all suck to high heaven, better just move on to the next girl.
Theoretically, you might still stand a chance with her once she sees the girl that was smart enough to agree to dance with you having the time of her life:) But whatever you do, don't stand there and aggravate her by arguing with her as to why she won't dance with you. Forget her, move on, there's a better-looking and smarter girl (for she knows to appreciate your invitation:) right behind her, and YOU are what she's been waiting for all her life:) Or at least this evening:)
How about when the girl you want to dance with is already dancing with somebody else? Here's a trick by Vilius which might just work, especially if she is bored with him and he is only a clueless chump. Don't use it on a really good friend though, that just wouldn't be right. But as for the rest, here goes: "First of all you must be with a good sense of humour and not be afraid to be "bad". You see your friend or an acquaintance (or any other clueless chump, doesn't really matter) dancing with the chick you want. Just come to the pair and interrupt them by taking over the MALE partner and just start to dance with him. Usually, the chick won't go anywhere and it'll take just a couple of seconds to feel the rhytm. After some 30 seconds you can just turn around, grab the girl, gently pull her a few feet away and continue dancing with her. Easy, funny trick;)".
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