15.6. References 651
LetTnbe the number of different combinations ofnmishaps that Dan can suffer
in one day (where we regard different sequences ofbreaksas different combina-
tions). For example,T 0 DT 1 D 0 , since there are always two or more breaks. On
the other hand,T 3 D 36 ; the reasoning is that there can be three breaks (which can
happen in 33 D 27 different combinations), or a spill and two breaks (which can
happen in 32 D 9 different combinations).
(a)Express the generating function
T.x/WWDT 0 CT 1 xCT 2 x^2 CT 3 x^3 C
as a quotient of polynomials andexplain your derivation.
(b)T.x/can be written in the form
T.x/D
9x
2
1
1 3x