Probability made me realise that...

Fine afternoon. I was looking expectantly at my older cousin. He was looking disdainfully at his younger cousin(me).A fat S.N.Dey book lay propped up against a big jar of cookies. The circumstances were quite favourable for what i was doing-learning probability.
"So, probability..." he began.
"I know the basic thing", i said "favourable occurrences by possible occurrences."
"Nicely put.Now, why don't you show me some real stuff? Do this." he said pointing triumphantly at a problem. He always believes me when i say that i am slow though i try to pass it for sarcasm.I had decided ages ago that he isn't too smart because no one takes my sarcasm seriously.
problem: A bag contains 6 red and 4 white balls.If two balls are taken out what is the probability that both of them are white?
There are some days when i am at my dumbest, not that i am at my smartest in the other days but on these days i have difficulty understanding practically everything. And a number of incidents since the morning had proved that it was one such day.
"Wait a second" i said, the rusty old wheels suddenly turning in my head with surprising alacrity. That was quick.
total possibilities= 10!/8!*2!=45
favourable possibilities= 6!/2!*4!=15
therefore probability= 15/45= 1/3

"Piece of cake", i said, feeling intensely relieved. I just escaped from a barrage of his choicest of curses.
"Okay", he said, obviously pleased because i had detained him for the past 3 hours,"If you do this you are done and if you cant do this, well, then i wont teach you any further because its futile".

"All right!" i said cheerfully, looking at the next problem. Then it struck me. There was something wrong with that statement. Either way he stops teaching me and escapes. That wasn't right! Still...
There are 6 chairs. If 4 boys and 2 girls are arranged randomly, what is the probability that the two girls will be seated together?
"Easy" i said overconfidently.
total possibilities=6!
favourable possibilities=5!
therefore the answer= 5!/6!= 1/6
"Done!" i said, smirking.
He looked at my work and shook his head.
"You moron, cant the girls interchange their position?" he scoffed. It was only a probability problem but his look suggested that i had confused a typewriter for a palmtop.
"Oh!",my eyes went perfectly round.
"The answer is= 5!*2/6!= 1/3", he said.
"But in the last problem, the one with balls i didn't...oh!" i exclaimed.
"what?" he asked warily.
"Balls and girls are different!" I said excitedly.
"what a find bro!", he said, laughing.
"O!"