At the law firm Minland Charles Dingle, Mahoney Charles Arnt, and Krencher Chester Clute learn that Judy Jones is entitled to inherit her late uncle's $10 million fortune on one condition: she must marry an intellectual genius within two weeks. When the firm cannot locate the real Judy Jones, they launch a campus-wide advertising campaign to find her, offering a nominal reward of $374.26.
The real Judy Jones Julie Bishop works as a nightclub singer for her boyfriend Tommy Coles's Robert Alda big-band radio program. Tommy believes she is the genuine Judy, and he remembers that her uncle was an archaeologist. The next day, Judy arrives at the firm with an antique shrunken head and family-history details that the attorneys verify as accurate, confirming that she might indeed be the heiress in question.
To secure the fortune, Judy decides to enroll at an all-male elite university, using a pseudonym. She is quickly discovered, but Professor Popik S.Z. Sakall allows her to complete the exam because of her striking appearance. Although she clearly fails the test, Popik convinces the college board of trustees to keep her enrolled, arguing that the fortune could fund a new chemistry laboratory. When she moves into her dorm, Professor Bart Williams William Prince tells Judy she is ineligible to attend, but he changes his mind when he learns how her wealth could transform the school's facilities.
To stay on campus, Judy takes a waitress job at the campus coffee shop. When Bart places an order, Judy accidentally serves him a sandwich with a bar of soap. In retaliation, Bart bluntly tells her that she represents everything he hates about women. Tommy arrives to defend Judy, but Bart knocks him unconscious, a moment that strangely impresses Judy and stirs the possibility of romance.
Popik grows convinced that Judy and Bart’s friction could bloom into love. He arranges for Tommy's band to perform on campus, hoping the public performances will bring the two closer together. Yet Bart accidentally eats another soap-soap sandwich and hurries away, once again knocking out Tommy. Later, Popik is arrested on suspicion of being an axe murderer; he escapes from prison with Bart's help and is later acquitted at trial.
As the marriage deadline nears, Judy invites Tommy's band to a charity bazaar near a lakeshore resort, and Popik teams up with the lawyers gathered there to push Judy and Bart toward each other. Mr. Keating [Edward Everett Horton] rows Judy into the middle of the lake and deliberately overturns the boat, prompting a rescue by Williams. In the struggle, Bart proposes to Judy, and the moment feels right enough to pursue a lifelong partnership.
During the drive to Nevada to be married, a police officer stops the group for speeding. In a jail cell, Tommy reveals that he graduated from law school and was once a quiz kid, and Judy is impressed by his legal knowledge, which tilts her feelings toward him. Meanwhile, the authorities arrest the real axe murderer, clearing the air around the couples. The next morning they are released, and though their drive is interrupted by a parade of returning soldiers, a military chaplain officiates the marriage between Judy and Tommy, sealing a union built from wit, misdirection, and a shared sense of humor in the face of a quite unconventional path to a happy ending.
