312 6 Combinatorics and Probability
world situation, namely that about particles and states. The above considerations apply
to bosons, particles that obey the Bose–Einstein statistics, which allows several particles
to occupy the same state. Examples of bosons are photons, gluons, and the helium-4
atom. Electrons and protons, on the other hand, are fermions. They are subject to the
Pauli exclusion principle: at most one can occupy a certain state. As such, fermions obey
what is called the Fermi–Dirac statistics.
A third problem comes from C. Reischer, A. Sâmboan,Collection of Problems
in Probability Theory and Mathematical Statistics(Editura Didactica ̧ ̆si Pedagogic ̆a,
Bucharest, 1972). It shows how probabilities can be used to prove combinatorial identi-
ties.
Example.Prove the identity
1 +
n
m+n− 1+···+
n(n− 1 )··· 1
(m+n− 1 )(m+n− 2 )···m=
m+n
m.
Solution.Consider a box containingnwhite balls andmblack balls. LetAibe the event
of extracting the first white ball at theith extraction. We compute
P(A 1 )=
m
m+n,
P(A 2 )=
n
m+n·
m
m+n− 1,
P(A 3 )=
n
m+n·
n− 1
m+n− 1·
m
m+n− 2,
···
P(Am)=n
m+n·
n− 1
m+n− 1···
1
m.
The eventsA 1 ,A 2 ,A 3 ,...are disjoint, and therefore
1 =P(A 1 )+P(A 2 )+···+P(Am)=m
m+n[
1 +
n
m+n− 1+···+
n(n− 1 )··· 1
(m+n− 1 )(m+n− 2 )··· 1]
.
The identity follows.
Because it will be needed below, let us recall that the expected value of an experiment
with possible outcomesa 1 ,a 2 ,...,anis the weighted mean
a 1 P(X=a 1 )+a 2 P(X=a 2 )+···+anP(X=an).So let us see the problems.