## shubhamsrg 3 years ago if [x] denotes greatest integer < or = x , then prove that C(n,3) - [ n/3 ] is a natural no. , divisible by 3 for all integer n> or =3

1. shubhamsrg

C(n,3) here denotes comination, for e.g. C(4,3) = 4! / (4-3)! (3)! likewise..

2. experimentX

|dw:1345190132115:dw||dw:1345190209186:dw| n can have only 3 from ... 1) 3k 2) 3k+1 3) 3k+2

3. shubhamsrg

am all years..what next ?

4. shubhamsrg

lol..i meant "ears"

5. experimentX

|dw:1345190372362:dw| this is just a try man ... i never know the solution beforehand.

6. shubhamsrg

that 4 wont be there in the last step...anyways nice... it'll be always divisible by sine its 9*something..and one of k and k-1 is always even so the 2 cancelled out..thus 3k satisfies it.. nice..

7. shubhamsrg

i meant it'll always be divisible by 3**

8. experimentX

lol ... how ... you always need k of the form to be k, k - 1, k - 2

9. experimentX

Oh ... sorry ... such a huge error!! :(((

10. experimentX

same thing repeats in other cases .. since you have three terms of m, m+1, m+2 <--- one of them is going to be divisible by 3

11. experimentX

and that [n/3] thing is not going to change.

12. shubhamsrg

excellent,,thans a lot :)

13. experimentX

not really ... i guess i should get some more sleep :)

14. shubhamsrg

yeah you must be tired..go and relax..even very sharp minds need sound sleep! ;)