Thread: Combinatorics: Generating functions

1. Find the generating function for the ways of distributing n loonies to 5 people so that the first two people have together either at most 8 loonies or an even number.

2. How many 9 digit numbers are there whose sum of digits is equal to 24?

For #1, I know how to find the generating function if the first two people have at most 8 loonies together AND an even number. But I am not sure how to approach this problem.

For #2, isn't this equivalent to asking how many solutions there are to to x1+x2+...+x9=24 with x1=>1? My solution is as follows: C(23+9-1, 9-1) - C(23+8-1, 8-1) = C(31,8) - C(30-7) = 5,852,925. But apparently the answer is 5,949,615.

Any help would be appreciated.

Hey BrownianMan.

Do you need to come up with an analytic expression or can you use a computer?

Originally Posted by BrownianMan

1. Find the generating function for the ways of distributing n loonies to 5 people so that the first two people have together either at most 8 loonies or an even number.
[/SIZE][/FONT][/SIZE][/FONT][FONT=CMR10][SIZE=3][FONT=CMR10][SIZE=3]For #1, I know how to find the generating function if the first two people have at most 8 loonies together AND an even number. But I am not sure how to approach this problem.

You say that you know how the do it for well .
I would like to see your solution.

Originally Posted by BrownianMan

How many 9 digit numbers are there whose sum of digits is equal to 24?

Assuming the first digit is not zero the coefficient of in
gives the answer.

See this.

Originally Posted by chiro

Hey BrownianMan.

Do you need to come up with an analytic expression or can you use a computer?

I can use a computer.

Originally Posted by Plato

You say that you know how the do it for well .
I would like to see your solution.


Assuming the first digit is not zero the coefficient of in
gives the answer.

See this.

Ok, I got #1. Thank you.

For #2, I am not sure how you got that expression? I know it gives the right answer, but could you explain it a bit?

Originally Posted by BrownianMan

For #2, I am not sure how you got that expression? I know it gives the right answer, but could you explain it a bit?

The first factor counts the non-zero digits for the lead digit. The next factor counts any digit for the last eight positions in the nine digit number.