Question

a+b+c+d=k
a,b,c,d>=1
can you find a formula for the number of integer solutions as a function of k?

Answer 1

It looks like repeated choose operation.

Answer 2

\[\binom{k-1}{3}\]

Answer 3

how did u get that so fast lol

Answer 4

it is a good ol stars and bars problem

\(k\) stars and \(3\) bars

Answer 5

wait wont u have a 0 in the middle with that form

Answer 6

oh did u subtract 3 spots

Answer 7

for the 3 zeros not allowed

Answer 8

\[\star\star|\star\star|\star|\star\star\]

Answer 9

like if zeros are allowed it would be K+3 choose 3 right

Answer 10

Yep

Answer 11

how come this works, so u subtract 4 spots? for the 4 zeros no allowed

Answer 12

Firstly, notice that those 3 bars divide the 7 stars into 4 pieces

Answer 13

right

Answer 14

You want to have at least 1 star in each piece, yes ?

Answer 15

right

Answer 16

okay i see now that limits the total number of places

Answer 17

ahhh

Answer 18

thats it!
so only the places between stars are allowed

Answer 19

dang that was simple lol

Answer 20

i got to this point with some other little trick lol

Answer 21

like u know if u look at the difference of the solutions that has a stars and bar solution again allowing 0

Answer 22

solving a+b+c+d = k over nonnegative integers ?

Answer 23

kinds cool that exists

Answer 24

yeah like umm

Answer 25

say we were looking at a+b+c=7
then the simple case
1+1+5=7
now i looked at their differences