If \(G\) is a grammar with productions \(S\rightarrow SaS\mid aSb\mid bSa\mid…

Key insight: every derivation preserves the property that the number of a's is at least the number of b's.

Reason:

• ε has #a − #b = 0.

• If x and y are derivable with differences dx and dy, then S → S S gives difference dx + dy (addition preserves nonnegativity).

• S → S a S adds one a, increasing the difference by 1.

• S → a S b and S → b S a add one a and one b, so they do not change the difference.

Thus every string generated satisfies #a − #b ≥ 0. The string babba has 2 a's and 3 b's (difference −1), so it cannot be generated by the grammar. The other listed strings have at least as many a's as b's and can be constructed (see individual option feedback for short derivations).