Question

ncr/ncr-1=n-r+1/r
prove this​

Answer 1

Answer:

nCr = n! / (n-r)! x r! and r! = r x (r-1)!

nCr

= n! / (n-r)! x r!

= n! / (n-r)! x r(r-1)!

nCr-1

= n! / (n-(r-1))! x (r-1)!

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

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

(n+1)Cr

= (n+1)! / ((n+1) – r)! x r!

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

now

LHS

= nCr + (n+1)Cr

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

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

= n! ( n-r+1-r / (n+1 – r)(n-r)! x r! )

= n! ( n+1 / ((n+1) – r)! x r! )

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

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

= (n+1) C r

= RHS