求证:(1)A(n+1,n+1)-A(n,n)=n^2A(n-1,n-1); (2)C(m,n+1)=C(m-1,n)+C(m,n-1)+C(m-1,n-1)求证:(1)A(n+1上标,n+1下标)-A(n上标,n下标)=n^2A(n-1上标,n-1下标)(2)C(m上标,n+1下标)=C(m-1上标,n下标)+C(m上标,n-1下标)+C(m-1上标,n-1下标)
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![求证:(1)A(n+1,n+1)-A(n,n)=n^2A(n-1,n-1); (2)C(m,n+1)=C(m-1,n)+C(m,n-1)+C(m-1,n-1)求证:(1)A(n+1上标,n+1下标)-A(n上标,n下标)=n^2A(n-1上标,n-1下标)(2)C(m上标,n+1下标)=C(m-1上标,n下标)+C(m上标,n-1下标)+C(m-1上标,n-1下标)](/uploads/image/z/11663101-37-1.jpg?t=%E6%B1%82%E8%AF%81%3A%281%29A%28n%2B1%2Cn%2B1%29-A%28n%2Cn%29%3Dn%5E2A%28n-1%2Cn-1%29%3B+%282%29C%28m%2Cn%2B1%29%3DC%28m-1%2Cn%29%2BC%28m%2Cn-1%29%2BC%28m-1%2Cn-1%29%E6%B1%82%E8%AF%81%3A%281%29A%28n%2B1%E4%B8%8A%E6%A0%87%2Cn%2B1%E4%B8%8B%E6%A0%87%29-A%28n%E4%B8%8A%E6%A0%87%2Cn%E4%B8%8B%E6%A0%87%29%3Dn%5E2A%28n-1%E4%B8%8A%E6%A0%87%2Cn-1%E4%B8%8B%E6%A0%87%29%282%29C%28m%E4%B8%8A%E6%A0%87%2Cn%2B1%E4%B8%8B%E6%A0%87%29%3DC%28m-1%E4%B8%8A%E6%A0%87%2Cn%E4%B8%8B%E6%A0%87%29%2BC%28m%E4%B8%8A%E6%A0%87%2Cn-1%E4%B8%8B%E6%A0%87%29%2BC%28m-1%E4%B8%8A%E6%A0%87%2Cn-1%E4%B8%8B%E6%A0%87%29)
求证:(1)A(n+1,n+1)-A(n,n)=n^2A(n-1,n-1); (2)C(m,n+1)=C(m-1,n)+C(m,n-1)+C(m-1,n-1)求证:(1)A(n+1上标,n+1下标)-A(n上标,n下标)=n^2A(n-1上标,n-1下标)(2)C(m上标,n+1下标)=C(m-1上标,n下标)+C(m上标,n-1下标)+C(m-1上标,n-1下标)
求证:(1)A(n+1,n+1)-A(n,n)=n^2A(n-1,n-1); (2)C(m,n+1)=C(m-1,n)+C(m,n-1)+C(m-1,n-1)
求证:(1)A(n+1上标,n+1下标)-A(n上标,n下标)=n^2A(n-1上标,n-1下标)
(2)C(m上标,n+1下标)=C(m-1上标,n下标)+C(m上标,n-1下标)+C(m-1上标,n-1下标)
排列组合
求证:(1)A(n+1,n+1)-A(n,n)=n^2A(n-1,n-1); (2)C(m,n+1)=C(m-1,n)+C(m,n-1)+C(m-1,n-1)求证:(1)A(n+1上标,n+1下标)-A(n上标,n下标)=n^2A(n-1上标,n-1下标)(2)C(m上标,n+1下标)=C(m-1上标,n下标)+C(m上标,n-1下标)+C(m-1上标,n-1下标)
(1)A(n+1,n+1) = (n+1)!= (n+1)*n*...*2*1
所以题目左边 = (n+1)!-(n)!= (n+1-1)*(n)!= (n*n)*(n-1)!= 右边,得证
(2)把右边的每个数都写成C(m,n) = n!/(m!*(n-m)!)的形式,
右边(字母太多看着也烦,就不列了)通分成分母为(m!*(n-m+1)!)的形式
右边 = ( m*n!+ (n-m)*(n-m+1)*(n-1)!+ (n-m+1)*m*(n-1)!))/(m!*(n-m+1)!)
= ( (n+1)!)/(m!*(n-m+1)!)
= 左边
命题得证.