iks84im62a的部落格

標題:

我有兩條證明題唔識做，唔該曬！！

發問:

1) (n+1)Ck=(nC(k-1))+(nCk) 2) (nCr)(rCk)=(nCk)((n-k)C(r-k))

最佳解答:

• I.S. question
• G.E.M.20point!!!!!!!!!!!@1@
• lim(x→0)(x^x^x) = -@1@

 

此文章來自奇摩知識+如有不便請留言告知

Note: nCr = n! / r!(n - r)! 又, Factorial: n! = n(n - 1)(n - 2)...(1) = n(n - 1)! (n + 1)! = (n + 1)n(n - 1)...(1) = (n + 1)n! 等等 1) R.H.S. = nC(k - 1) + nCk = n!/(k - 1)![n - (k - 1)]! + n!/k!(n - k)! = n!/(k - 1)!(n - k + 1)! + n!/k!(n - k)! 通分母 = kn!/k(k - 1)!(n - k + 1)! + (n - k + 1)n!/k!(n - k + 1)(n - k)! = kn!/k!(n - k + 1)! + (n - k + 1)n!/k!(n - k + 1)! = [kn! + (n - k + 1)n!] / k!(n - k + 1)! = (k + n - k + 1)n! / k![(n + 1) - k]! = (n + 1)n! / k![(n + 1) - k]! = (n + 1)! / k![(n + 1) - k]! = (n + 1)Ck = L.H.S. 2) R.H.S. = (nCk)[(n - k)C(r - k)] = [n!/k!(n - k)!]{(n - k)!/(r - k)![(n - k) - (r - k)]!} 約掉(n - k)! = n!/k!(r - k)!(n - k - r + k)! = n!/k!(r - k)!(n - r)! = [n!/(n - r)!][1/k!(r - k)!] 分子和分母乘以r! = [n!/r!(n - r)!][r!/k!(r - k)!] = (nCr)(rCk) = L.H.S.

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