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Joint Entrance Examination
Mathematics
Binomial Theorem
Question 114
Question 114 of 347
Medium
MCQ
The value
∑
r
=
0
22
(
22
r
)
(
23
r
)
\displaystyle\sum_{r=0}^{22}\binom{22}{r}\binom{23}{r}
r
=
0
∑
22
(
r
22
)
(
r
23
)
is
A
45
C
23
{}^{45}C_{23}
45
C
23
B
44
C
23
{}^{44}C_{23}
44
C
23
C
45
C
24
{}^{45}C_{24}
45
C
24
D
44
C
22
{}^{44}C_{22}
44
C
22
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Official solution
\[\sum_{r=0}^{22}\binom{22}{r}\binom{23}{r} = \sum_{r=0}^{22}\binom{22}{r}\binom{23}{23-r} = \binom{45}{23}\quad(\text{Vandermonde's identity})\]
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