Two balls, having linear momenta $$\overrightarrow{p}_1 = p\hat{i}$$ and $$\overrightarrow{p}_1 = -p\hat{i}$$, undergo a collision in free space. There is no external force acting on the balls. Let $$\overrightarrow{p}_1'$$ and $$\overrightarrow{p}_2'$$ be their final momenta. The following option(s) is(are) NOT ALLOWED for any non-zero value of $$p, a_1, a_2, b_1, b_2, c_1$$ and $$c_2$$.

$$\overrightarrow{p}_1' = a_1\hat{i} + b_1\hat{j} + c_1\hat{k}$$
$$\overrightarrow{p}_2' = a_2\hat{i} + b_2\hat{j}$$

$$\overrightarrow{p}_1' = c_1\hat{k}$$
$$\overrightarrow{p}_2' = c_2\hat{k}$$

$$\overrightarrow{p}_1' = a_1\hat{i} + b_1\hat{j} + c_1\hat{k}$$
$$\overrightarrow{p}_2' = a_2\hat{i} + b_2\hat{j} - c_1\hat{k}$$

$$\overrightarrow{p}_1' = a_1\hat{i} + b_1\hat{j}$$
$$\overrightarrow{p}_2' = a_2\hat{i} + b_1\hat{j}$$