Recall the classification of vitamins

Vitamins are broadly classified into:

• Fat-soluble vitamins: $A, D, E, K$
• Water-soluble vitamins: B-complex vitamins and vitamin $C$

Check each vitamin given in the question

• A. Vitamin $B_1$: This is a B-complex vitamin, so it is water-soluble.
• B. Vitamin $C$: This is water-soluble.
• C. Vitamin $E$: This is fat-soluble.
• D. Vitamin $B_{12}$: This is also a B-complex vitamin, so it is water-soluble.
• E. Vitamin $K$: This is fat-soluble.

Identify the fat-soluble vitamins from the list

The fat-soluble vitamins are: $\boxed{\text{Vitamin }E \text{ and Vitamin }K}$

Match with the given options

• Option A: $B$ and $C$ only $\to$ Vitamin $C$ and Vitamin $E$ ❌
• Option B: $C$ and $D$ only $\to$ Vitamin $E$ and Vitamin $B_{12}$ ❌
• Option C: $C$ and $E$ only $\to$ Vitamin $E$ and Vitamin $K$ ✅
• Option D: $A$ and $B$ only $\to$ Vitamin $B_1$ and Vitamin $C$ ❌

Final answer

$\boxed{\text{Option C: } C \text{ and } E \text{ only}}$

Comparison with stored correct answer

Stored correct answer = C

My derived answer = C

Therefore, they agree.