Prove that r( n Pr-1) +n Pr = n+1 P r ......with steps
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Answer:
L.H.S. =
n
P
r
=
(n−r)!
n!
..(1)
R.H.S. =
n−1
P
r
+r.
n−1
P
r−1
=
(n−r−1)!
(n−1)!
+
(n−r)!
(n−1)!
=
(n−r−1)!
(n−1)!
[1+
n−r
r
]
=
(n−1).(n−r−1)!
n.(n−1)!
=
(n−r)!
n!
..(2)
Hence from (1) and (2)
n
P
r
=
n−1
P
r
+r.
n−1
P
r−1
.
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