Understanding the Question: Choosing the Right Pronoun

The question asks us to select the most appropriate option to fill in the blank in the sentence: "It is their problem. Let _____ solve it themselves." We need to choose the correct pronoun that fits grammatically and logically with the rest of the sentence, particularly in relation to the possessive pronoun "their" and the reflexive pronoun "themselves".

Analyzing the Sentence Structure and Pronoun Use

The sentence consists of two parts. The first part, "It is their problem," establishes that the problem belongs to a group of people (indicated by the possessive pronoun "their").

The second part, "Let  _____ solve it themselves," instructs or suggests that this same group should solve the problem. The word "Let" in this construction is often followed by an object pronoun (or a noun phrase) and a base form of the verb (solve). The reflexive pronoun "themselves" further confirms that the action of solving is performed by the subject of the verb "solve" and reflects back on that subject.

Since "their" refers to a plural group, the pronoun filling the blank must also refer to this same plural group and function as the object after "Let". The pronoun must agree in number with "their" and "themselves".

Evaluating the Options for the Blank

Let's look at the given options and see which one fits:

1. her: This is a singular, feminine object pronoun. It refers to a single female person. This does not match the plural "their" and "themselves".
2. it: This is a singular, neuter object pronoun, usually referring to a thing or animal. It does not match "their" and "themselves", which refer to people.
3. us: This is a plural object pronoun referring to 'we' (the speaker and others). The sentence starts with "It is their problem," indicating the problem belongs to 'them', not 'us'. It does not match "their" and "themselves".
4. them: This is the plural object pronoun corresponding to the subject pronoun "they". It refers to a group of people. This pronoun agrees with "their" and "themselves" in number (plural) and refers to the same group of people mentioned earlier.

Determining the Correct Object Pronoun

The structure "Let + object pronoun + base verb" is common for expressing permission or suggestion. The subject of the action (solving) is implied to be the object pronoun following "Let". Since the possessive pronoun "their" and the reflexive pronoun "themselves" clearly refer to a plural group, the object pronoun must also be plural and refer to this group. The correct object pronoun for a plural group is "them".

Filling the blank with "them" gives the sentence: "It is their problem. Let them solve it themselves." This sentence is grammatically correct and makes perfect sense, indicating that the group whose problem it is should be allowed or encouraged to solve it on their own.

Conclusion: Selecting 'Them'

Based on the analysis of pronoun agreement and sentence structure, the most appropriate pronoun to fill the blank is "them". It correctly refers to the plural group indicated by "their" and is the correct object pronoun to follow "Let" in this construction, aligning with the reflexive pronoun "themselves".

Revision Table: Key Pronoun Types

Additional Information: The 'Let + Object + Verb' Construction

The construction "Let + object pronoun/noun phrase + base form of the verb" is used for several purposes in English grammar:

• To give permission: "Let me go."
• To make a suggestion (when using "us"): "Let's go" (from "Let us go").
• To express a request or command (less common, often sounds formal or like an instruction): "Let the children play outside."
• In clauses following "Let": "Let them solve it." Here, "them" is the object pronoun, and "solve" is the base verb. This structure indicates who is supposed to perform the action of solving.

In the context of the original sentence, "Let them solve it themselves," the structure clearly assigns the action of "solve" to "them", who are the people referred to by "their" and "themselves".