Could you please explain to me how to complete this truth table?
I think this problem looks more complicated than it actually is due to the extra qualifying "information" supplied.

In order for Y to be high it is necessary for either QA, and/or both the QB, QC pair to be switched off (high-impedance state). This is because QA is in series with the QB, QC pair thus:

QA           QB and QC   Y

OFF          Don't Care  HIGH
Don't Care   OFF         HIGH
ON           ON          LOW


Since QB and QC are in parallel the combination of these can only be off if both QB and QC are off. So overall the truth table is:

    QA    QB    QC    Y

    OFF   OFF   OFF   HIGH
    OFF   ON    OFF   HIGH
    OFF   OFF   ON    HIGH
    OFF   ON    ON    HIGH

    ON    OFF   OFF   HIGH
    ON    ON    OFF   LOW
    ON    OFF   ON    LOW
    ON    ON    ON    LOW