As soon as the word "paki" was heard, everyone jumped straight to the racism conclusion without really thinking about it. This isn't surprising, as much modern commentary is of the speak first, think later variety, but this particular happening also demonstrates the superficial thinking that surrounds the issue of racism.
Racism nowadays is so strongly verboten that even the slightest tinge is enough to have people running for cover. The result of this is lack of proper, sensible debate and the consequence of no debate is that we now have no idea of where the boundaries lie between out and out racism (ie bigotry) and fair comment and freedom of speech. So instead of leaping to the worst possible conclusion about Prince Harry, lets just apply a little thought.
The first thing to think about is context. In fact context is so important, its worth saying again:
What is the context?
In this case it was several things: a privately made video set in an extremely close knit group. Setting aside the issue of privacy (who reading this would be happy for their every private comment to appear in the national press?), the key issue is the relationship between Prince Harry and his fellow Officer Cadets.
The course at Sandhurst is a year long, it's arduous and demanding both physically and mentally. This should be no surprise as the end result is the Queen's Commission, and we do hope these are not given away lightly, or given away at all. The point being, that such long and arduous courses lead to a level of camaraderie that is uncommon outside military life - other examples might be the better sports teams, arctic explorers perhaps and so forth. In that context, words and phrases that might otherwise have negative connotations can lose all their baggage and either revert to their basic meaning or perhaps take on new meaning all together. So in the case of Prince Harry, its quite possible that referring to a fellow cadet as a Paki was not even harmless but actually affectionate. After all "Paki" is just the diminutive of Pakistani, and we do know that the cadet so referred to was from Pakistan.
And that's just to consider the immediate context. The wider context is more illuminating yet; Sandhurst is there to train and prepare men to operate as leaders and commanders in the harshest of environments, to do this the cadets need to be tested in equally harsh environments. One way to generate a harsh environment is to use a degree of verbal abuse. To anyone thinking that this would be inappropriate I'd say, what do you propose instead? If a cadet messes up in training he gets an unpleasant talking to amongst other things; if he messes up on operations he may end up dead, or, possibly worse, someone else ends up dead! If you can't handle a little bit of the verbal, what are you going to be like when the shooting starts? As it happens we know a couple of things about the Pakistani cadet: he wasn't offended and he can handle worse than a bit of verbal.
And what do you think Prince Harry was called during his training when he messed up? Do you imagine that someone might, just might, have referred to the colour of his hair, maybe? Is it out of all likelihood that he was ever referred to as a "ginger tosser" perhaps, maybe with a smile and a laugh, or possibly with greater venom if the occasion demanded it.
Taken out of context, such things can sound offensive or worse, but that isn't the point. You have to judge words in their context, what's more if you're not yourself familiar with the context perhaps you should make no judgement at all.