Two $4$ bits binary numbers, $A = 1101$ and $B = 1010$ are given in the inputs of a logic circuit shown in figure below. The output $(Y)$ will be:

From the given logic circuit, the input $A$ passes through a NOT gate, giving $\overline{A}$.

This output $\overline{A}$ and the input $B$ are then fed into a NAND gate.

The Boolean expression for the final output $Y$ is:
$Y = \overline{\overline{A} \cdot B}$

Given the $4$-bit binary inputs:
$A = 1101$
$B = 1010$

First, find the bitwise NOT of $A$:
$\overline{A} = \overline{1101} = 0010$

Next, find the bitwise AND of $\overline{A}$ and $B$:
$\overline{A} \cdot B = 0010 \cdot 1010 = 0010$

Finally, apply the NOT operation for the NAND gate:
$Y = \overline{0010} = 1101$

Alternatively, using De Morgan's laws:
$Y = \overline{\overline{A} \cdot B} = A + \overline{B}$
$Y = 1101 + \overline{1010} = 1101 + 0101 = 1101$

Therefore, the output $Y$ is $1101$.

Answer: $Y = 1101$