Explain solution to problem 2 of AP physics Sample Questions repost

I am reposting this question as I am still unclear on the answer I got – 

Here’s the problem:

See this link for College Board Link to CB Sample questions. Go to PDF 161 or page 154. Question begins: “The diagram above shows a top view of a child of mass M…”

I think that answers A and C are wrong because moving the child in or out changes the velocity of the system (conservation of momentum?)  but I do not understand about rotational inertia. Is there a formula to help in examining this problem?

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