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Mathematics
Mathematics
Problem
Let
m
,
n
,
r
m,n,r
m
,
n
,
r
be non-negative integers with
0
≤
r
≤
m
+
n
0\le r\le m+n
0
≤
r
≤
m
+
n
. a. (i) Evaluate numerically, for
m
=
4
,
n
=
3
,
r
=
3
m=4,\;n=3,\;r=3
m
=
4
,
n
=
3
,
r
=
3
, the sum
∑
k
=
0
3
(
4
k
)
(
3
3
−
k
)
\sum_{k=0}^{3}{4\choose k}{3\choose 3-k}
k
=
0
∑
3
(
k
4
)
(
3
−
k
3
)
and compare it with
(
7
3
)
{7\choose 3}
(
3
7
)
. b. (ii) Prove, for general
m
,
n
,
r
m,n,r
m
,
n
,
r
, the identity
∑
k
=
0
r
(
m
k
)
(
n
r
−
k
)
=
(
m
+
n
r
)
.
\sum_{k=0}^{r}{m\choose k}{n\choose r-k}={m+n\choose r}.
k
=
0
∑
r
(
k
m
)
(
r
−
k
n
)
=
(
r
m
+
n
)
.
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