2 replies to this topic

[mmiguel]

Find a closed-form expression for F(a,b) where:

F(a,b) = F(a-1,b) + F(a-1,b-1),
F(a,0) = 1 for all a
F(0,b) = 0 for all b

a and b are positive integers

[Anza Power]

What does F(0,0) evaluate to? you need it for F(1,1)

Spoiler for

In case F(0,0)=1 then F(a,b) = b choose a, as in the binomial coefficient between b and a.

In case F(0,0)=0 then it's the binomial coefficient between b-1 and a.

Given F(0,0) there is only one solution, you can see this if you draw an xy axis and mark a on the x axis and b on the y axis, then mark 0's on the a axis and 1's on the b axis, then the value of every other point on the plane is the sum of values of the point to it's left and the point under that...

Edited by Anza Power, 21 October 2012 - 03:04 PM.

[mmiguel]

What does F(0,0) evaluate to? you need it for F(1,1)

Spoiler for

In case F(0,0)=1 then F(a,b) = b choose a, as in the binomial coefficient between b and a.

In case F(0,0)=0 then it's the binomial coefficient between b-1 and a.

Given F(0,0) there is only one solution, you can see this if you draw an xy axis and mark a on the x axis and b on the y axis, then mark 0's on the a axis and 1's on the b axis, then the value of every other point on the plane is the sum of values of the point to it's left and the point under that...

F(0,0) = 1
Sorry for leaving that out.

Nice work, you are correct!