The number of non negative integral solutions of `3x+y+z=24` is

Explanation

We have,
`3x+y+z=24` and given `xge0,yge0,zge0`
let x=k
`thereforey+z=24-3k` . . . (i)
here, `24ge24-3kge0[becausex ge0]`
hence, `0 le k le 8`
The total number of integral solutions of Eq. (i) is
`.^(24-3k+2-1)C_(2-1)=.^(25-3k)C_(1)=25-3k`
Hence, the total number of sols of the original equation
`=underset(k=0)overset(8)(sum)(25-3k)=25underset(k=0)overset(8)(sum)1-3underset(k=0)overset(8)(sum)=k`
`=25*9-3*(8-9)/(2)=225-108=117`