Concrete mathematics : a foundation for computer science

168 BINOMIAL COEFFICIENTS
Equation (5.21) holds primarily because of cancellation between m!‘s in
the factorial representations of (A) and (T) . If all variables are integers and
r 3 m 3 k 3 0, we have
>( >
r m
=--
T! m!
m k m!(r-m)! k!(m-k)!
r.I
=k!
(m- k)! (r-m)!
= -- r! (?.--I! = (;)(;;“k>.
k!(r-k)! (m-k)!(r-m)!
That was easy. Furthermore, if m < k or k < 0, both sides of (5.21) are Yeah, right.
zero; so the identity holds for all integers m and k. Finally, the polynomial
argument extends its validity to all real r.
A binomial coefficient 1:;) = r!/(r - k)! k! can be written in the form
(a + b)!/a! b! after a suitab1.e renaming of variables. Similarly, the quantity
in the middle of the derivation above, r!/k! (m - k)! (r - m)!, can be written
in the form (a + b + ~)!/a! b! c!. This is a “trinomial coefficient :’ which arises
in the “trinomial theorem” :
(a+b+c)!
(x+y+z)n = t a! b! c!
O$a,b,c
“Excogitavi autem
olim mirabilem
regulam pro numeris
coefficientibus
potestatum, non
So (A) (T) is really a trinomial coefficient in disguise. Trinomial coefficients
pop up occasionally in applications, and we can conveniently write them as
tanturn a bhomio
x + y , sed et a
trinomio x + y + 2,
(a + b + c)!
imo a polynomio
quocunque, ut data
(aaTE,Tc) = a!b!
potentia gradus
cujuscunque v.
in order to emphasize the symmetry present.
Binomial and trinomial coefficients generalize to multinomial coefi-
gr. decimi, et
potentia in ejus
valore comprehensa,
bents, which are always expressible as products of binomial coefficients: ut x5y3z2, possim
statim assignare
al + a2 + . . . + a,
al,a2,...,a, >
_= (al +az+...+a,)!
al ! ar! . . . a,!
numerum coefficientem,
quem
habere debet, sine
==
a1 + a2 + . . . + a,
a2 + . . . + a, > “’ (““h,‘am) .
ulla Tabula jam
calculata
--G.,V~~ibni~[~()fJ/
Therefore, when we run across such a beastie, our standard techniques apply.