Alice can complete a certain work in 24 days. Bob is twice as efficient as Alice. Both of them worked together for x days and stopped. The remaining work was completed by Carol working alone for (x+1) days. If Carol is 25 % less efficient than Bob, then the total number of days it took to complete the whole work was

Alice completes the work in 24 days, so her work rate = $\frac{1}{24}$ of the work per day.

Bob is twice as efficient as Alice, so Bob's work rate = $2 \times \frac{1}{24} = \frac{1}{12}$ of the work per day.

Carol is 25% less efficient than Bob. This means Carol works at 75% of Bob's efficiency.

Carol's work rate = $0.75 \times \frac{1}{12} = \frac{3}{4} \times \frac{1}{12} = \frac{1}{16}$ of the work per day.

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When Alice and Bob work together for $x$ days:

Combined work rate = $\frac{1}{24} + \frac{1}{12} = \frac{1}{24} + \frac{2}{24} = \frac{3}{24} = \frac{1}{8}$ of the work per day.

Work completed by Alice and Bob = $\frac{x}{8}$

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Carol then works alone for $(x+1)$ days:

Work completed by Carol = $\frac{x+1}{16}$

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Since the entire work is completed:

$\frac{x}{8} + \frac{x+1}{16} = 1$

Multiplying throughout by 16:

$2x + (x+1) = 16$

$3x + 1 = 16$

$3x = 15$

$x = 5$

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Total days to complete the work = Days Alice and Bob worked + Days Carol worked

Total days = $x + (x+1) = 5 + 6 = \mathbf{11}$ days

The answer is Option 3: 11.