Finding non-trivial sufficient statistic

Question

I have this question.

To find a sufficient I have to find first the joint pmf of this pmf's?

Let $X_{1}, . . . , X_{n} $be a random sample from the following pmf: $P(X = k_{1}) = \frac{1 − \theta}{2}$ , $P(X = k_{2}) = \frac{1}{2}$ and $P(X = k_{3}) = \frac{\theta}{2}$ , $0 < \theta < 1$ Find a non-trivial sufficient statistic for $\theta$.

Answer 1

$$L(\theta)=\left(\frac{1-\theta}{2} \right)^{\sum_{i=1}^n 1_{k_1}(x_i)} \left(\frac{\theta}{2} \right)^{\sum_{i=1}^n 1_{k_3}(x_i)} \left(\frac{1}{2} \right)^{n-\sum_{i=1}^n 1_{k_1}(x_i)-\sum_{i=1}^n 1_{k_3}(x_i)} $$ $\left(\sum_{i=1}^n 1_{k_1}(x_i),\sum_{i=1}^n 1_{k_3}(x_i)\right)$ is a sufficient statistic.