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Putnam
1987 Putnam
B2
B2
Part of
1987 Putnam
Problems
(1)
Putnam 1987 B2
Source:
8/5/2019
Let
r
,
s
r,s
r
,
s
and
t
t
t
be integers with
0
≤
r
0 \leq r
0
≤
r
,
0
≤
s
0 \leq s
0
≤
s
and
r
+
s
≤
t
r+s \leq t
r
+
s
≤
t
. Prove that
(
s
0
)
(
t
r
)
+
(
s
1
)
(
t
r
+
1
)
+
⋯
+
(
s
s
)
(
t
r
+
s
)
=
t
+
1
(
t
+
1
−
s
)
(
t
−
s
r
)
.
\frac{\binom s0}{\binom tr} + \frac{\binom s1}{\binom{t}{r+1}} + \cdots + \frac{\binom ss}{\binom{t}{r+s}} = \frac{t+1}{(t+1-s)\binom{t-s}{r}}.
(
r
t
)
(
0
s
)
+
(
r
+
1
t
)
(
1
s
)
+
⋯
+
(
r
+
s
t
)
(
s
s
)
=
(
t
+
1
−
s
)
(
r
t
−
s
)
t
+
1
.
Putnam