Concept:
Polarity of an organic compound depends upon the magnitude of charge separation within the molecule. The greater the difference in electronegativity between bonded atoms and the greater the resultant dipole moment, the more polar the molecule becomes.
The polarity of common oxygen-containing organic compounds generally follows the trend:
\[
\text{Ether} \lt \text{Ketone} \lt \text{Alcohol} \lt \text{Carboxylic Acid}
\]
This trend arises because alcohols and carboxylic acids are capable of extensive intermolecular hydrogen bonding, while ethers and ketones are comparatively less polar.
Step 1: Examine compound A \((CH_3CH_2OCH_2CH_3)\).
Compound A is diethyl ether.
Although oxygen is electronegative and creates a dipole in the C--O bonds, the molecule contains only one oxygen atom and no O--H bond.
Therefore, hydrogen bonding is absent and the overall polarity is comparatively low.
Hence, ether is the least polar among the given compounds.
\[
A = CH_3CH_2OCH_2CH_3
\]
Step 2: Examine compound C \((CH_3COCH_3)\).
Compound C is acetone, a ketone.
The carbonyl group \((C=O)\) possesses a strong dipole due to the large electronegativity difference between carbon and oxygen.
Therefore, ketones are more polar than ethers.
\[
A \lt C
\]
Step 3: Examine compound B \((CH_3CH_2OH)\).
Compound B is ethanol.
The presence of the O--H bond makes alcohols capable of intermolecular hydrogen bonding.
Hydrogen bonding significantly increases polarity compared to ketones.
Therefore,
\[
C \lt B
\]
Step 4: Examine compound D \((CH_3COOH)\).
Compound D is acetic acid.
Carboxylic acids contain both a carbonyl group \((C=O)\) and a hydroxyl group \((O-H)\).
As a result, they possess very strong intermolecular hydrogen bonding and exist as dimers.
This makes them the most polar among the given compounds.
Hence,
\[
B \lt D
\]
Step 5: Write the final increasing order.
Combining all the comparisons obtained above:
\[
A \lt C \lt B \lt D
\]
Thus, the correct option is
\[
\boxed{(1)\; A \lt C \lt B \lt D}
\]