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Joint Entrance Examination
Mathematics
Permutations and Combinations
Question 249
Question 249 of 278
Medium
MCQ
If
n
C
r
^nC_r
n
C
r
denotes the number of combinations of
n
n
n
things taken
r
r
r
at a time, then the expression
n
C
r
+
1
+
n
C
r
−
1
+
2
⋅
n
C
r
^nC_{r+1}+{}^nC_{r-1}+2\cdot{}^nC_r
n
C
r
+
1
+
n
C
r
−
1
+
2
⋅
n
C
r
equals:
A
n
+
1
C
r
+
1
^{n+1}C_{r+1}
n
+
1
C
r
+
1
B
n
+
2
C
r
^{n+2}C_r
n
+
2
C
r
C
n
+
2
C
r
+
1
^{n+2}C_{r+1}
n
+
2
C
r
+
1
D
n
+
1
C
r
^{n+1}C_r
n
+
1
C
r
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Official solution
\[^nC_{r+1}+{}^nC_{r-1}+2\cdot{}^nC_r = ({}^nC_{r-1}+{}^nC_r) + ({}^nC_r+{}^nC_{r+1})\] \[= {}^{n+1}C_r + {}^{n+1}C_{r+1} = {}^{n+2}C_{r+1}\]
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