Consider the following statements:

(A) Propanal and Propanone are functional isomers

(B) Ethoxyethane and methoxypropane are metamers

(C) But-2-ene shows optical isomerism

(D) But-1-ene and But-2-ene are functional isomers

(E) Pentane and 2,2-dimethylpropane are chain isomers

The correct statements are

To determine the correct statements, let us analyze each of the given statements step-by-step:

Statement (A): Propanal and Propanone are functional isomers
Propanal is an aldehyde with the structural formula:
${\mathrm{CH}}_3{\mathrm{CH}}_2\mathrm{CHO}$
Propanone is a ketone with the structural formula:
${\mathrm{CH}}_3{\mathrm{COCH}}_3$
Both compounds share the same molecular formula, $C_3{H}_{6}O$, but contain different functional groups (aldehyde vs. ketone). Therefore, they are functional isomers. Thus, statement (A) is correct.

Statement (B): Ethoxyethane and methoxypropane are metamers
Ethoxyethane is a symmetric ether with the formula:
${\mathrm{CH}}_3{\mathrm{CH}}_2-O-{\mathrm{CH}}_2{\mathrm{CH}}_3$
Methoxypropane is an unsymmetric ether with the formula:
${\mathrm{CH}}_3-O-{\mathrm{CH}}_2{\mathrm{CH}}_2{\mathrm{CH}}_3$
Metamers are isomers that have the same molecular formula ($C_4{H}_{10}O$) but differ in the alkyl groups attached to the divalent heteroatom (oxygen). Since the alkyl groups on either side of the oxygen atom are different (ethyl/ethyl vs. methyl/propyl), these two molecules are metamers. Thus, statement (B) is correct.

Statement (C): But-2-ene shows optical isomerism
But-2-ene has the chemical structure:
${\mathrm{CH}}_3-\mathrm{CH}=\mathrm{CH}-{\mathrm{CH}}_3$
While it exhibits geometrical (cis-trans) isomerism due to restricted rotation around the double bond, it does not possess any chiral (asymmetric) carbon atom. Without a chiral center or overall molecular asymmetry, it cannot exhibit optical isomerism. Thus, statement (C) is incorrect.

Statement (D): But-1-ene and But-2-ene are functional isomers
Both But-1-ene and But-2-ene are alkenes with the molecular formula:
$C_4{H}_8$
Because they belong to the same homologous series and contain the same functional group (a carbon-carbon double bond), they are not functional isomers. Instead, they differ only in the position of the double bond (carbon-1 vs. carbon-2), making them position isomers. Thus, statement (D) is incorrect.

Statement (E): Pentane and 2,2-dimethylpropane are chain isomers
Pentane is a straight-chain alkane:
${\mathrm{CH}}_3{\mathrm{CH}}_2{\mathrm{CH}}_2{\mathrm{CH}}_2{\mathrm{CH}}_3$
2,2-dimethylpropane is a branched-chain alkane:
${\left({\mathrm{CH}}_3\right)}_{4}C$
Both compounds share the same molecular formula, $C_5{H}_{12}$, but differ in the arrangement of their carbon skeleton (straight chain vs. highly branched chain). Thus, they are chain isomers. Thus, statement (E) is correct.

In summary, the correct statements are (A), (B), and (E).