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Jozsef Wildt International Math Competition
2009 Jozsef Wildt International Math Competition
W. 8
Prove this combinatorial formula
Prove this combinatorial formula
Source: 2009 József Wildt International Mathematical Competition
April 17, 2020
combinatorics
algebra
Problem Statement
If
n
,
p
,
q
∈
N
,
p
<
q
n,p,q \in \mathbb{N}, p<q
n
,
p
,
q
∈
N
,
p
<
q
then
(
(
p
+
q
)
n
n
)
∑
k
=
0
n
(
−
1
)
k
(
n
k
)
(
(
p
+
q
−
1
)
n
p
n
−
k
)
=
(
(
p
+
q
)
n
p
n
)
∑
k
=
0
[
n
2
]
(
−
1
)
k
(
p
n
k
)
(
(
q
−
p
)
n
n
−
2
k
)
{{(p+q)n}\choose{n}} \sum \limits_{k=0}^n (-1)^k {{n}\choose{k}} {{(p+q-1)n}\choose{pn-k}}= {{(p+q)n}\choose{pn}} \sum \limits_{k=0}^{\left [\frac{n}{2} \right ]} (-1)^k {{pn}\choose{k}} {{(q-p)n}\choose{n-2k}}
(
n
(
p
+
q
)
n
)
k
=
0
∑
n
(
−
1
)
k
(
k
n
)
(
p
n
−
k
(
p
+
q
−
1
)
n
)
=
(
p
n
(
p
+
q
)
n
)
k
=
0
∑
[
2
n
]
(
−
1
)
k
(
k
p
n
)
(
n
−
2
k
(
q
−
p
)
n
)
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