FabR[Senior Patek Moderator]
26166
My pleasure, and the math is correct! ;-)) Here's a simpler way to compute that sum and arrive at 32 (I'm writing it quickly as I'm sure you'll understand it given the math you already wrote ;-) It follows from the following small theorem/observation:
Jul 01, 2021,20:57 PM
Observation: In your notation, the sum c(N,0) + c(N,1) + ... + c(N,N) equals 2^N ("2 raised to the N"), for any N.
(Begin proof: Any given element either belongs to a subset or it doesn't, giving you two choices per element. Hence, the total number of choices, or of subsets, is 2 x 2 x ... x 2 (N times)=2^N. End of proof ;-)
The situation FPJ was studying for his watch was simply the special case corresponding to N=5 (the number of fingers), yielding a desired total of 2^5=32
Cheers.