For the really simple case, if you have N characters (a b c d e f ..) and want to make them into pairs, you have N choices for the first half of the pair, and then N-1 choices for the second (assuming you can’t choose the first choice twice; if you can, then you have N choices the second time too). If the pair “a b” is the same as the pair “b a”, then you have to divide the number of possible pairs in half to account for this double-counting. So probably the answer you want is N*(N-1) / 2, but depending the specific details you may get a different answer. (One other case I’m glossing over is what if you have duplicates in your set, like a a a b b b – you’ll have to decide whether that matters or not.)

“After a song or two, and before she could reply to the entreaties of several that she would sing again, she was eagerly succeeded at the instrument by her sister Mary, who having, in consequence of being the only plain one in the family, worked hard for knowledge and accomplishments, was always impatient for display.

Mary had neither genius nor taste; and though vanity had given her application, it had given her likewise a pedantic air and conceited manner, which would have injured a higher degree of excellence than she had reached. Elizabeth, easy and unaffected, had been listened to with much more pleasure, though not playing half so well”

This basically lists three separate components: genius, taste, and application. My read of this is that “application” here means “practices hard”, “genius” means “has a natural flair for performing the work”, and “taste” means “is able to distinguish the *product* as being good or bad, including things that she produces herself as well as things that other people produces”.

The implication to me is that Elizabeth has applied herself less but has better taste, which enables her to produce a better product because she knows her own limitations; Mary misestimates how good her playing is, so she tries things that are beyond her skill and sounds bad.