Over the past week, Trump had made obvious his annoyance that twisting Attorney General Jeff Sessions’s arm won’t do anything. Sessions recused himself from decision-making related to the Russia investigation after it was revealed that he offered untrue testimony during his Senate confirmation hearings about meetings with the Russian ambassador. So the task fell to Sessions’s No. 2, deputy attorney general Rod J. Rosenstein. It was Rosenstein who appointed Mueller, to Trump’s annoyance. To oust Mueller, Trump needs someone new as Rosenstein’s boss, meaning that he needs Sessions gone.

Weirdly, Trump hasn’t simply fired Sessions, choosing, instead, to apparently try to humiliate Sessions into quitting. (Firing an attorney general is not without precedent. Harry Truman did it, though under very different circumstances.) As it turns out, there may be a reason for that.

Paul Butler is a professor of law at Georgetown University. In an interview with The Post on Thursday, he explained the various ways in which Trump could replace Sessions, if Sessions resigned or were fired.

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There are four.

1. Trump nominates a new attorney general who is confirmed by the Senate. This is the normal route that would be used by a president in most circumstances. It’s also a route that would face opposition from even Republican members of the Senate, as indicated by a warning tweet from the Senate Judiciary Committee Chairman Charles E. Grassley. His committee would need to consider such an appointment; Grassley says there’s “no way” that happens this year.

2. Trump lets Rosenstein take over. As noted above, this is not Trump’s preferred recourse, since Rosenstein appointed Mueller and could fire him now if he wanted to. (And if he had cause: The provisions of the special counsel statute dictate that there must be a valid reason for firing Mueller once appointed.)

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3. Trump can appoint someone while the Senate is recessed. The Founding Fathers recognized that there may be a need for a president to fill a Cabinet position when the Senate is unable to confirm someone for the position. (Say your secretary of war dies while the Senate is back home in the 13 original states.) So the president can appoint someone while the Senate is recessed, awaiting later confirmation.

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This power has been abused. President Barack Obama’s recess appointment of members of the National Labor Relations Board was thrown out by the Supreme Court in 2014, because the Senate had been holding pro forma sessions specifically to prevent such appointments at the time Obama made them. Democrats in the Senate have made clear that they plan to do everything in their power to similarly block Trump from making such an appointment.

That leaves the fourth option:

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4. The Vacancies Reform Act of 1998. This was the heart of what Butler discussed with us.

“The Vacancies Act says that — notwithstanding any kind of other rule, like the regular DOJ succession statute — that the president, and only the president, can appoint an acting head of the Department of Justice to that position as long as that person has been confirmed by the Senate,” Butler explained. “So if there’s another person who works in the federal government whose job requires presidential appointment and then confirmation by the Senate, then that person is entitled to be put in, in this case, as the acting attorney general.”

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So, in other words, if Sessions were to resign, President Trump could appoint, say, Rick Perry, the secretary of energy, to be the acting attorney general? “The answer is yes,” Butler said, since Perry has already been appointed by Trump and confirmed by the Senate.

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But there’s a catch.

“The statute says that it applies when the current officeholder ‘dies, resigns or is otherwise unable to perform the functions and duties of the office,’” he said. “That suggests that there might be a difference and the Act might not apply if the person is fired.” So if Trump gets Sessions to quit, Rick Perry could be made acting attorney general. If Sessions is fired? It’s less clear.

Butler noted that the debate over the act included this question. Some senators thought that it should be interpreted to apply in the case of a firing, but it’s not clear. Were Trump to fire Sessions and then try to replace him under the Vacancies Act, it would likely end up being settled by the courts.

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There’s another question, too. If Perry is moved into Justice to fire Mueller — could he? Technically, Rosenstein now supervises Mueller. In most circumstances, he’d unquestionably surrender that supervision to a new superior, should one be appointed. In this case? He might be more hesitant to do so. But, Butler figures, he’d still do so — in part because there are looming questions about whether or not Rosenstein might at some point need to recuse himself, given his having written a letter to Trump that was used to justify the firing of FBI Director James B. Comey — a firing that itself has now been reportedly looped into Mueller’s investigation.

“The bottom line is that if Trump wants to put his own person in place, a person who would have supervisory authority over the special counsel and have the ability to dismiss the special counsel, then this Vacancies Act is his best route,” Butler said.