Washington (CNN) -- Mitt Romney is in a bind over binders. The Republican presidential nominee is being challenged about the details of hiring women for top government positions when he won election as governor of Massachusetts in 2002, giving a possible late boost to efforts by President Barack Obama's campaign to strengthen its support among female voters. Less than three weeks before Election Day, the battle for the women's vote -- a demographic Obama won easily in 2008 and needs again this year to secure re-election -- appeared to be tightening. Then comments by Romney in Tuesday's second of three presidential debates raised questions about what actually happened when he was governor and provided an opening for Obama's team to challenge whether he understands the role of women in the workplace and the economy. "I had the chance to pull together a cabinet and all the applicants seemed to be men," Romney said at the debate in answer to a question about equal pay for women. "And I went to my staff, and I said, 'How come all the people for these jobs are all men?' They said, 'Well, these are the people that have the qualifications.' And I said, 'Well, gosh, can't we find some women that are also qualified?'" That led to a "concerted effort to go out and find women who had backgrounds that could be qualified to become members" of his cabinet, he continued, saying: "I went to a number of women's groups and said, 'Can you help us find folks?' And they brought us whole binders full of women." Romney added that a university study later concluded that Massachusetts under his leadership "had more women in senior leadership positions than any other state in America." The "binders full of women" reference promoted a huge response on social media, becoming the third-fastest rising Google search during the debate and getting a Twitter hashtag, a series of memes on Tumblr and a Facebook page with more than 8,000 members by Thursday morning.