Binomial (1 Viewer)

[OLDMAN]

Got a new online student courtesy of bored, must not offend God(laz) and contribute some.

The idea for this question is from a discussion in a 3-unit thread.


Prove without using formulas :
nCk = n-1Ck-1 + n-1Ck

 

[KeypadSDM]

Ok, no formulas. But then what can we use? The very ideas behind probability define the formula, so if we work from first principles, we end up using formulas anyway.

Pascals triangle maybe, but that in itself isn't proven.

 

[turtle_2468]

Soln follows, look away ***SPOILER WARNING*** if you want to do it yourself.

Case bash.
Consider n objects, and no. of ways of choosing k of them. Let one of them arbitrarily be called A.
Case 1: A is chosen. Then we have to choose k-1 from the other n-1, hence n-1C(k-1).
Case 2: A isn't chosen, then similarly (n-1)Ck.
Done, as one of these cases always works (ie they can't both work and exactly 1 must work)

 

[KeypadSDM]

So simple and elegant. I never thought of splitting it up like that.

 

[OLDMAN]

Hello Dr. Maths!

 

[ND]

This question came up when i had tutoring with spice girl, he had the same answer.

 

[spice girl]

i think turtle and i both learnt it off ben burton in our first maths camp...wot, bak in dec '99?

 

[OLDMAN]

Goes to show that a good mathematician needs to have the memory of an elephant - elephantician, to add to mathmagician. Build up your math memory bank!

Actually, my inspiration for this question is from humbler sources Jones/Couchman's old 3-unit book:
"Give a verbal explanation why nCr = nCn-r."

 

[KeypadSDM]

Originally posted by spice girl
i think turtle and i both learnt it off ben burton in our first maths camp...wot, bak in dec '99?

Ben Burton? Serious. He was so pissed off at me by the end of NMSS 2003.