Question

If x, y, z are integers and $x\ge 0,y\ge 1,z\ge 2,x+y+z<15$ then the number of values of the ordered triplet $(x,y,z)$ is

• A. 91
• B. 455
• C. ${}^{17}{C}_{15}$
• D. none of these

Answer 1

The correct option is A 91

Substituting $x=0$, and $y=1$ , we get 13 ways.
Hence 13 ordered pairs.
Substituting, $x=1$, and $y=1$ we get 12 ways .
Hence 12 ordered pairs.
So on...
Substituting $x=12$, and $y=1$, we get 1 way
Therefore the required number of ways is
$13+12+11+...1$
$\frac{13\left(14\right)}{2}$
$=13\times 7$
$=91$