## AravindG 3 years ago find the no: of integral solutions of inequation x+y+z+u<=65

1. FoolForMath

0 is included?

2. FoolForMath

I can do this using binomial coefficients.

3. AravindG

i dont know tht fool

4. FoolForMath

Using stars and bars, the answer is given by $\huge \sum \limits_{n=0}^{65} \binom{n+3}{3}$

5. satellite73

can't wait to see this

6. satellite73

stars and bars??

7. AravindG

hw?

8. FoolForMath
9. moneybird

this question is like putting three bookmarks in a book

10. satellite73

how cool is that? stars and bars, nice visual!

11. FoolForMath

Yeah it's coool :D

12. AravindG

coool

13. shaan_iitk

One can also think logically and get the solution .. say you have x+y+z+u <= k (some integer) and you have to find the number of solutions then it is like finding the number of ways where k sweets can be distributed to 4 kids and total sweets distributed may not be equal to k. Now lets imagine a situation where we have 4 women and k men. How many ways of arranging them? (4+k)!/(4!*k!) Now say I say that all men who come to the left of any woman are given to that woman (not literally) .. then we have solved our distribution problem and hence our equation.. so I believe the answer should be 69!/(4!*65!)

14. FoolForMath

@shaan_iitk How do you think stars and bars work? That is exactly the same thing you elaborated.

15. shaan_iitk

means?

16. shaan_iitk

Sorry but I am not aware of stars and bars ..

17. FoolForMath

I have posted the url before.

18. shaan_iitk

okk .. I didn't look at that.. I only looked at your expression which was in summation form.. I thought logically we don't need that summation..