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National and Regional Contests
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India LIMIT
2019 LIMIT
2019 LIMIT Category A
Problem 3
counting finite sequences
counting finite sequences
Source: LIMIT 2019 CAS2 P3
April 28, 2021
combinatorics
Problem Statement
How many finite sequences
x
1
,
x
2
,
…
,
x
m
x_1,x_2,\ldots,x_m
x
1
,
x
2
,
…
,
x
m
are there such that
x
i
=
1
x_i=1
x
i
=
1
or
2
2
2
and
∑
i
=
1
m
x
i
=
10
\sum_{i=1}^mx_i=10
∑
i
=
1
m
x
i
=
10
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
89
<span class='latex-bold'>(A)</span>~89
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
89
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
73
<span class='latex-bold'>(B)</span>~73
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
73
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
107
<span class='latex-bold'>(C)</span>~107
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
107
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
119
<span class='latex-bold'>(D)</span>~119
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
119
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