Question

The number of positive integral solutions of $x+y+z=n,nϵN,n\ge 3$, is

• A. ${}^{n-1}{C}_2$
• B. ${}^{n-1}{P}_2$
• C. $n(n-1)$
• D. none of these

Answer 1

Answer: (A)

${}^{n-1}{C}_2$

Consider $n=3$,
Hence we get
$x+y+z=3$
Since x,y,z all are non-zero positive integers.
$x=y=z=1$
Hence number of solutions is 1.
$={}^2{C}_2$
$={}^{3-1}{C}_2$
Consider n=4
$x+y+z=4$
$x=1,y=1,z=2$
$x=2,y=1,z=1$
$x=1,y=2,z=1$
Hence there are 3 solutions.
$={}^3{C}_2$
$={}^{4-1}{C}_2$
Hence for $x+y+x=n$ and $n\ge 3$
The number of positive integral solutions is ${}^{n-1}{C}_2$