F.4 Maths Factorial Notation

By using C(n,r) + C(n,r-1) = C(n+1,r) :C(n,r) + C(n,r-1) + C(n+1,r-1) + C(n+2,r-1) + ... + C(n+k,r-1)
=.......... C(n+1,r) + C(n+1,r-1) + C(n+2,r-1) + ... + C(n+k,r-1)
= ............................C(n+2 , r) + C(n+2,r-1) + ... + C(n+k,r-1)
= ............................................. C(n+3,r) + C(n+3,r-1) + ... + C(n+k,r-1)
= ...........................
= C(n+k , r) + C(n+k , r-1) + C(n+k,r-1)
= C(n+k+1 , r)i.e.

C(n,r-1) + C(n+1,r-1) + C(n+2,r-1) + ... + C(n+k,r-1) = C(n+k+1,r) - C(n,r)

2012-11-15 20:31:44 補充：
The 7th line :
= C(n+k , r) + C(n+k , r-1) + C(n+k,r-1)

should be
= C(n+k , r) + C(n+k , r-1)

2012-11-15 20:37:54 補充：
Alternatively :

∵ nCr = (n+1)C(r+1) - nC(r+1)

∴
nC(r-1) + (n+1)C(r-1) + (n+2)C(r-1) + ... + (n+k)C(r-1)


= [(n+1)Cr - nCr ] + [(n+2)Cr - (n+1)Cr] + [(n+3)Cr - (n+2)Cr] + ... + [(n+k)Cr - (n+k-1)Cr] + [ (n+k+1)Cr - (n+k)Cr]


= (n+k+1)Cr - nCr

2012-11-15 20:48:35 補充：
The proof of nC(r-1) + nCr = (n+1)Cr :

nC(r-1) + nCr

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

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

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

2012-11-15 20:48:41 補充：
= n! (n + 1) / [r! (n - r + 1)!]

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

= (n+1)Cr

You should remember it~