Question

Prove That
p(n+1,r)=(n+1)p(n,r-1)​

Answer 1

$\huge\sf\underline{\red{Answer}}$

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

.