popper-logic-scientific-discovery

APPENDIX iii
Derivation of the First Form of
the Binomial Formula (for finite
sequences of overlapping segments,
cf. section 56)
The first binomial formula* 1
(1)
α F″(m) = (n) nCmpm n − m q
where p = αF″(1), q = αF″(0), m � n, may be said to be proved under the
assumption that α is (at least) n − I-free (neglecting errors arising at the
last term; cf. section 56), if we can show that
(2)
α F″(σ (n) m) = pm n − m q
where ‘σ m’ denotes one particular n-tuple (although an arbitrarily
chosen one) which contains m ones. (The symbol is intended to indicate
that what is given is the complete arrangement of this n-tuple, i.e.
not only the number of ones but also their positions in the n-tuple.) For
* 1 Note that ( n m) is an alternative way of writing the binomial coefficient n C m, i.e. the
number of ways in which m things may be arranged in n places, provided m � n.