1. The problem states that the number of permutations of $n$ objects taken 2 at a time is 12, i.e., $_nP_2 = 12$. 2. The formula for permutations of $n$ objects taken $r$ at a time is: $$ _nP_r = \frac{n!}{(n-r)!} $$ 3. Here, $r=2$, so: $$ _nP_2 = \frac{n!}{(n-2)!} = 12 $$ 4. Expanding factorials for $r=2$: $$ \frac{n!}{(n-2)!} = n \times (n-1) = 12 $$ 5. This gives the quadratic equation: $$ n(n-1) = 12 $$ 6. Simplify: $$ n^2 - n - 12 = 0 $$ 7. Factor the quadratic: $$ (n-4)(n+3) = 0 $$ 8. The solutions are: $$ n = 4 \quad \text{or} \quad n = -3 $$ 9. Since $n$ represents the number of objects, it must be positive, so: $$ n = 4 $$ Final answer: $n=4$.