The balls, having linear momenta $\vec{p}_1 = p\hat{i}$ and $\vec{p}_2 = -p\hat{i}$, undergo a collision in free space. There is no external force acting on the balls. Let $\vec{p}_1'$ and $\vec{p}_2'$ 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 $c_2$.

Step 1: Determine the initial total momentum of the system.

Since there is no external force acting on the balls, the total linear momentum of the system is conserved.

$$\vec{p}_1 = p\hat{i}$$ $$\vec{p}_2 = -p\hat{i}$$ $$\vec{P}_{\text{initial}} = \vec{p}_1 + \vec{p}_2 = 0$$

Step 2: Apply the conservation of linear momentum.

The total final momentum must equal the total initial momentum.

$$\vec{P}_{\text{final}} = \vec{p}_1' + \vec{p}_2' = 0$$ $$\vec{p}_2' = -\vec{p}_1'$$

Step 3: Analyze Option (1).

The final momenta are given as:

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

Calculate the total final momentum:

$$\vec{P}_{\text{final}} = (a_1 + a_2)\hat{i} + (b_1 + b_2)\hat{j} + c_1\hat{k}$$

Since $c_1 \neq 0$, the $\hat{k}$ component of the total momentum can never be zero, violating momentum conservation.

Step 4: Analyze Option (2).

The final momenta are given as:

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

Calculate the total final momentum:

$$\vec{P}_{\text{final}} = (c_1 + c_2)\hat{k}$$

Choosing $c_2 = -c_1$ yields $\vec{P}_{\text{final}} = 0$, which is allowed.

Step 5: Analyze Option (3).

The final momenta are given as:

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

Calculate the total final momentum:

$$\vec{P}_{\text{final}} = (a_1 + a_2)\hat{i} + (b_1 + b_2)\hat{j}$$

Choosing $a_2 = -a_1$ and $b_2 = -b_1$ yields $\vec{P}_{\text{final}} = 0$, which is allowed.

Step 6: Analyze Option (4).

The final momenta are given as:

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

Calculate the total final momentum:

$$\vec{P}_{\text{final}} = (a_1 + a_2)\hat{i} + 2b_1\hat{j}$$

Since $b_1 \neq 0$, the $\hat{j}$ component of the total momentum can never be zero, violating momentum conservation.

Answer: Options (1) and (4) are NOT ALLOWED.