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Putnam
1962 Putnam
B1
B1
Part of
1962 Putnam
Problems
(1)
Putnam 1962 B1
Source: Putnam 1962
5/21/2022
Let
x
(
n
)
=
x
(
x
−
1
)
⋯
(
x
−
n
+
1
)
x^{(n)}=x(x-1)\cdots (x-n+1)
x
(
n
)
=
x
(
x
−
1
)
⋯
(
x
−
n
+
1
)
for
n
n
n
a positive integer and let
x
(
0
)
=
1.
x^{(0)}=1.
x
(
0
)
=
1.
Prove that
(
x
+
y
)
(
n
)
=
∑
k
=
0
n
(
n
k
)
x
(
k
)
y
(
n
−
k
)
.
(x+y)^{(n)}= \sum_{k=0}^{n} \binom{n}{k} x^{(k)} y^{(n-k)}.
(
x
+
y
)
(
n
)
=
k
=
0
∑
n
(
k
n
)
x
(
k
)
y
(
n
−
k
)
.
Putnam
binomial theorem
Binomial