數學的排列與組合

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


2010-12-08 23:12:10 補充：
nCr 公式是 n! / [(r!) (n - r)!] ,

所以 nC(r-1)

= n! / { (r - 1)! [n - (r - 1)] ! }

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

中間通分母 , 一定要意識到 (r - 1)! * r = r! ,
及 (n - r)! * (n - r + 1) = (n - r + 1)! , (這些都是根據階乘原理 : m! * (m+1) = (m+1) !)

於是通分母以 [r! (n - r + 1)!] 為目標。

2010-12-08 23:12:15 補充：
最後一步是掉轉用公式 n! / [(r!) (n - r)!] = nCr ,

所以 (n+1)! / [r! (n+1 - r)!] = (n+1)Cr

2010-12-09 11:52:53 補充：
第4個=中的n! (n + 1)

怎麼做到第5個=中的(n + 1)!

階乘原理 : n! * (n+1) = (n+1) !

例如 :

4! * 5
= 1*2*3*4*5
= 5!

99! * 100
= 1*2*3*...*98*99*100
= 100!

nCr-1+nCr
=n!/((r-1)!(n-(r-1))!)+n!/(r!(n-r)!)
=n!/((r-1)!(n-r+1)(n-r)!))+n!/(r(r-1)!(n-r)!)
=(rn!+n!(n-r+1))/(r(r-1)!(n-r+1)(n-r)!)
=n!(r+n-r+1)/(r!((n+1)-r))
=(n+1)!/(r!((n+1)-r))
=n+1Cr