Begin2.DVI
ben green
(Ben Green)
#1
Binomial Coefficients
The binomial expansion (p+q)n, for nan integer, can be expressed in the form
(p+q)n=
(
n
0
)
pn+
(
n
1
)
pn−^1 q+
(
n
2
)
pn−^2 q^2 +···+
(
n
m
)
pn−mqm+···+
(
n
n
)
qn (11 .35)
Note in the special case p=q= 1 one obtains
(1 + 1)n= 2n= 1 +
∑n
m=1
(
n
m
)
= 1 +
(
n
1
)
+
(
n
2
)
+···+
(
n
n
)
(11 .36)
and rearranging terms one finds
∑n
m=1
(
n
m
)
=
(
n
1
)
+
(
n
2
)
+···+
(
n
n
)
= 2n− 1 (11 .37)
Let pdenote the probability that an event will happen and q= 1 −pdenote the
probability that the event will fail in a single trial and examine the probability that
the event will happen in n-trials. In one trial (p+q) = 1. In two trials, the expansion
(p+q)^2 =
(
2
0
)
p^2 +
(
2
1
)
pq +
(
2
2
)
q^2 =p^2 + 2pq +q^2
represents all possible outcomes. For example, if the trial is flipping a coin and pis
the probability of heads Hand qis the probability of tail T, then p^2 is the probability
of getting two successive heads HH , 2 pq is the probability of getting a head and tail
or tail and head HT or T H and q^2 is the probability of getting two tails T T. Similarly,
in three trials of flipping a coin the expansion
(p+q)^3 =
(
3
0
)
p^3 +
(
3
1
)
p^2 q+
(
3
2
)
pq^2 +
(
3
3
)
q^3
gives the probabilities
(
3
0
)
p^3 =p^3 is probability of getting three successive heads HHH
(
3
1
)
p^2 q=3 p^2 qis the probability of getting HHT or HT H or T HH
and represents the probability of getting two heads in 3 trials.
(
3
2
)
pq^2 =3 pq^2 is the probability of getting HT T or T HT or T T H
and represents the probability of getting one head in 3 trials.
(
3
3
)
q^3 =q^3 is the probability of getting T T T
and represents the probability of not getting a head in 3 trials.
