Show that n abcn1 a1bcn1 ab1cn1 abc1 for abcn N with ahcn Fo

Show that (n a,b,c)=(n-1 a-1,b,c)-(n-1 a,b-1,c)+(n-1 a,b,c-1) for a,b,c,n N with a+h+c=n. For extra credit, prove this in two distinct ways

Solution

Consider selecting a, b, c from n

This can be done in

n

a b c ways.

If 1 is already selected for a, then this is done in

n-1

a-1 b c ways

Thus if b is already selected 1, then

n-1

a b-1 c

Hence we can say

n = n-1 + n-1 + n-1

a b c a-1, b, c a, b-1, c a, b, c-1