The number of values of x for which $$C\left(\begin{array}{c}17-x\\ 3x + 1\end{array}\right)$$ is defined as an integer is

$$C\left(\begin{array}{c}17-x\\ 3x + 1\end{array}\right)$$ ; here the C refers to combination , ie nCr where n=17-x and r=3x+1

Now, we know nCr is defined as an integer when $$n\ge0,\ r\ge0\ and\ n\ge r$$ , where n and r are integers.

In the given expression, n=17-x, and r=3x+1

So, $$17-x\ge0;\ \ 3x+1\ge0\ and\ 17-x\ge3x+1$$

From 1st eqn, $$x\le17$$

From 2nd eqn, $$x\ge-\frac{1}{3}$$

And , from 3rd eqn, $$17-x\ge3x+1\ =>16\ge4x\ \ \ or,\ x\le4$$

Combining all these , we get one relation that satisfies all three inequalities, which is $$-\frac{1}{3}\le x\le4$$

And since 17-x and 3x+1 need to be integers, so x has to be an integer. The integers in this range are 0,1,2,3,4. 

So the number of values of x is 5.