summation help

Hey im having a little trouble with this:

S(nCm) = 2^n

So far ive wrote first few terms and have got that it will be 2+ something but can't figure out what the rest comes out to

You can prove this by induction.

Or do u need to actually need to deduce the 2^n result?

it says show that ""="" so im not sure - its from a uni prep problem sheet and im not sure proof by induction would be assumed knowledge?

I'd think so since its an A Level topic.

Even if you want to do this from scratch, you can test the sum for the first few values of n, notice powers of 2, conjecture that the sum is a power of 2 depending on n, then prove the result via induction. You would still show the result.

Thanks, do you think you could give me a hand with the next one too please?:

show that (N-1)C(r) + (N-1)C(r-1) = nCr, can I also just prove that by induction? I tried manipulating it but went wrong somewhere and got
LHS = (N+1)^2 * nCr

I wouldn't use induction on that one.

If you rewrite everything on the left in factorial format to start, then get everything over a common denominator (look to the RHS to see what denominator you're aiming for), it drops out in a couple of lines.

Post your working if it's still not coming out.

it says show that ""="" so im not sure - its from a uni prep problem sheet and im not sure proof by induction would be assumed knowledge?

A very short proof of this follows from considering the binomial expansion of (1+1)^n

show that (N-1)C(r) + (N-1)C(r-1) = nCr, can I also just prove that by induction? I tried manipulating it but went wrong somewhere and got
LHS = (N+1)^2 * nCr