Step 1: Understanding the Question:
We need to identify the pair of dissolved chlorides that are responsible for causing permanent hardness in water.
Step 2: Key Formula or Approach:
This question is based on the chemical definition of water hardness.
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Water Hardness: Caused by dissolved divalent cations, primarily Calcium (Ca\(^{2+}\)) and Magnesium (Mg\(^{2+}\)).
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Permanent Hardness: Caused by the chlorides and sulfates of these ions; this type is not removed by boiling.
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Temporary Hardness: Caused by the bicarbonates of these ions.
Step 3: Detailed Explanation:
Let's analyze the pairs in the options:
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(A) NaCl, KCl: Sodium and potassium ions are monovalent and do not cause hardness.
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(B) CaCl\(_2\), KCl: Calcium chloride (CaCl\(_2\)) causes permanent hardness, but potassium chloride (KCl) does not.
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(C) AlCl\(_3\), MgCl\(_2\): Magnesium chloride (MgCl\(_2\)) causes permanent hardness, but aluminum chloride is not considered a cause of common water hardness.
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(D) MgCl\(_2\), CaCl\(_2\): Both magnesium chloride and calcium chloride are the definitive salts responsible for permanent hardness.
Step 4: Final Answer:
The pair of chlorides responsible for permanent hardness of water is MgCl\(_2\), CaCl\(_2\).