Jim Blandings (Cary Grant), a bright account executive in the advertising business, lives with his wife Muriel (Myrna Loy) and two daughters, Betsy (Connie Marshall) and Joan (Sharyn Moffett), in a cramped New York apartment. Muriel secretly plans to remodel their apartment. After rejecting this idea, Jim Blandings comes across an ad for new homes in Connecticut and they get excited about moving. Planning to purchase and "fix up" an old home, the couple contact a real estate agent, who uses them to unload "The Old Hackett Place" in fictional Lansdale County, Connecticut. It is a dilapidated, two-hundred-year-old farmhouse. Blandings purchases the property for more than the going rate for land in the area, provoking his friend/lawyer Bill Cole (Melvyn Douglas) to chastise him for following his heart rather than his head.
The old house, dating from the Revolutionary War-era, turns out to be structurally unsound and has to be torn down. The Blandings hire architect Simms (Reginald Denny) to design and supervise the construction of the new home. From the original purchase to the new house's completion, a long litany of unforeseen troubles and setbacks beset the hapless Blandings and delay their moving-in date. On top of all this, at work Jim is assigned the task of coming up with a slogan for "WHAM" Brand Ham, an advertising account that has destroyed the careers of previous account executives assigned to it. Jim also suspects that Muriel is cheating on him with Bill Cole after Bill slept at the Blandings alone in the house with Muriel one night due to a violent thunderstorm.
With mounting pressure, skyrocketing expenses, and his new assignment, Jim starts to wonder why he wanted to live in the country. The Blandings maid Gussie provides Blandings with the perfect WHAM slogan, and he saves his job. As the film ends, Bill Cole says that he realizes that some things "you do buy with your heart."

Question: With whom does Jim believe Muriel cheated on him with?
Answer: Bill Cole
Alternative answer: Bill Cole