$\text{Consider the Euler variables of polyhedral objects namely P₁, P₂, P₃ and P₄ as given in table.}$

$\text{Which one of the following options is NOT a topologically valid closed polyhedral object as per Euler’s law?}$

Polyhedral object	Faces (F)	Edges (E)	Vertices (V)	Face inner loop (L)	Bodies (B)	Genus (G)
P₁	6	12	8	0	1	0
P₂	5	8	5	0	1	0
P₃	5	12	8	0	1	0
P₄	10	24	16	0	1	0

The correct option is

P₃

Step-by-Step Explanation:
According to Euler's formula for topologically valid closed polyhedral objects:

$$V-E+F-L=2(B-G)$$

Let's verify this formula for each of the objects given in the table:

1. For P₁:
LHS: $V-E+F-L=8-12+6-0=2$
RHS: $2(B-G)=2(1-0)=2$
Since LHS = RHS, P₁ is topologically valid.

2. For P₂:
LHS: $V-E+F-L=5-8+5-0=2$
RHS: $2(B-G)=2(1-0)=2$
Since LHS = RHS, P₂ is topologically valid.

3. For P₃:
LHS: $V-E+F-L=8-12+5-0=1$
RHS: $2(B-G)=2(1-0)=2$
Here, LHS ($1$) ≠ RHS ($2$). Thus, P₃ is NOT a topologically valid closed polyhedral object.

4. For P₄:
LHS: $V-E+F-L=16-24+10-0=2$
RHS: $2(B-G)=2(1-0)=2$
Since LHS = RHS, P₄ is topologically valid.