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“It seems much more plausible and is consistent with the whispering around town that [the neocons] need some fresh voices,” he said. “I think it is possible that this is a shot across the bow of Rand Paul and the most skeptical interventionist who are now ascendant in the Republican Party. They are a minority, but a growing one.”

Mr. Paul welcomed Ms. Cheney to the race by saying his first response to the news was to wonder whether she was “going to run in her home state of Virginia.”

Ms. Cheney, 46, spent most of her adult life inside Beltway before she moved with her husband and children into a home in Jackson Hole, Wyo., that she bought last year.

Some Republicans say Ms. Cheney chose to run in Wyoming because Virginia is a much bigger state where a campaign would be much more costly. It also is a tough swing state with a popular incumbent in Sen. Mark Warner, a Democrat.

“She is clearly part of the Washington political elite that hatched a plan in Northern Virginia to either run in Virginia or Wyoming, and they picked the state that they thought would be easier or less expensive,” said Liz Brimmer, a veteran Republican Party consultant in Wyoming and Enzi supporter.