Question

prove that:
$\frac{n!}{r!\times (n-r)!} + \frac{n!}{(r-1)! \times (n-r+1)!} = \frac{(n+1)!}{r!\times (n-r+1)!}$

Answer 1

Answer:

n!/r!*(n - r)! + n!/(r  -  1)!*(n - r + 1)! = ( n + 1)!/r!(n + 1 - r)!]

Step-by-step explanation:

Hi,

Consider n!/r!*(n - r)! + n!/(r  -  1)!*(n - r + 1)!

We know that r! = r*(r - 1)!

(n - r + 1)! = (n - r + 1)*(n - r)!

n!/r!*(n - r)! + n!/(r  -  1)!*(n - r + 1)!

= n!/r*(r - 1)!*(n - r)! + n!/(r  -  1)!*(n - r + 1)*(n - r)!

Taking n!/(r -1)!(n - r)! as common, we get

=  n!/(r -1)!(n - r)! [ 1/r + 1/(n - r + 1)]

= n!/(r -1)!(n - r)! [n - r + 1 + r]/r*(n - r + 1)

= [n!/r!(n + 1 - r)!]*[ n + 1]

= ( n + 1)!/r!(n + 1 - r)!]

= R.H.S

Hope, it helps !