Explanation:

Entropy is a measure of the randomness or disorder in a system. In an isolated system, the total entropy remains constant or increases with time, according to the second law of thermodynamics. However, it is possible for the entropy to decrease locally in a small part of the system as long as the overall change in entropy of the system is positive.

Local Decrease in Entropy:
When we say that entropy may decline locally somewhere within the isolated system, it means that in a small part of the system, the entropy can decrease. This implies that the arrangement of particles or molecules within that region becomes more ordered or less random.

Increased Entropy Elsewhere:
However, the decrease in entropy in one part of the system must be balanced by an increase in entropy somewhere else within the system. This is necessary to satisfy the second law of thermodynamics, which states that the total entropy of an isolated system always increases or remains constant.

Explanation of Option C:
Option C states that the local decrease in entropy must be balanced by increased entropy somewhere within the system. This statement is correct because the overall entropy of an isolated system cannot decrease. If there is a decrease in entropy in a certain region, there must be compensatory increases in entropy elsewhere to maintain the total entropy of the system.

Example:
For example, consider a container divided into two compartments by a partition. Initially, the particles are evenly distributed in both compartments, resulting in a high level of randomness or entropy. If the partition is removed, the particles will spread out and become more uniformly distributed throughout the container. This leads to an increase in the overall entropy of the system.

However, during the removal of the partition, there may be a temporary decrease in entropy in the region where the partition was located. This is because the particles in that region become more ordered as they move away from the partition. But this temporary decrease in entropy is balanced by the increase in entropy in the rest of the system, resulting in an overall increase in entropy.

Therefore, option C is the correct answer as it accurately explains that a local decrease in entropy within an isolated system must be balanced by increased entropy elsewhere within the system.