Question

数学排列组合：C(m,n+1):C(m,n):C(m-2,n)=4:2:1求m,n值.m右上n右下角

Answer 1

因为C(m,n+1)=C(m,n)+C(m-1,n)

[C(m,n+1)]/C(m,n)=[C(m,n)+C(m-1,n)]/C(m,n)

=1+C(m-1,n)/C(m,n)=4/2=2

所以C(m-1,n)=C(m,n)

所以（m-1）+m=2m-1=n;

所以

[C(m,n)]/[C(m-2,n)]

=[C(m,2m-1)]/C(m-2,2m-1)

=2/1

=2,

C(m,2m-1)=(2m-1)!/{m![(2m-1)-m]!}

=(2m-1)!/[m!(m-1)!]

C(m-2,2m-1)

=(2m-1)!/{(m-2)![(2m-1)-(m-2)]!}

=(2m-1)!/[(m-2)!(m+1)!]

所以

[C(m,n)]/[C(m-2,n)]

={(2m-1)!/[m!(m-1)!]}/{(2m-1)!/[(m-2)!(m+1)!]}

=[(m-2)!(m+1)!]/[m!(m-1)!]

=(m+1)/(m-1)

=2

所以

(m+1)/(m-1) =2

m=3

n=2m-1=5