Premier Giulio Andreotti, whose government fell Monday, was asked tonight to form another government and lead Italy out of its dangerous political crisis.

The invitation from President Giovanni Leone was widely predicted. But ANdreotti, a Christian Democrat like all the other premiers here since 1943, faces a long and difficult task.

His last government, the third he has headed, was brought down by the Communists, Italy's second biggest party. Andreotti is now expected to go back to those very same Communists to see if he can regain their support.

On paper, the Communists are demanding seats in a Christian Democratic Cabinet as the price for their support. The U.S. State Department has publicly declared Washington is opposition to any such deal.

In private, Communist leaders here make plain they will settle for less. But how much less of a share in decision-making and lower political profile they will accept is a question.

How much Andreotti can give is also in question. A strong and militant wing of his party opposes any new deal and wants an election.

But most politicians here, as everywhere, can think of dozens of reasons why they should not now face the voters. Among them is the fear that political terrorism, the worst in continental Europe, will be stepped up and that an election campaign will spark more killings, bombings and kidnapings.

Tonight at the Quirinale, the presidential palace, Andreotti told reporters he knew his minority Christian Democrats faced a hard job in trying to reform their government.

"I do not hide he difficulties," he said. "The positions of the parties are rather divergent."

He stressed that the first requirement of any new government is "to restore tranquility to all our cities," a reference to the wave of political violences. The second, he said, was finding more jobs, especially for the young, in a nation where unemployment is 8 per cent and rising.

But the premier-designate gave no hint of how much more of a share in decision-making he would offer the Communists or what common program they might agree on.

One shrewd Christian Democrat put Andreotti's problem this way: "He must invent a formula that lets us say Communists are not part of the majority and also enables the Communists to say there are."

This sort of trick sounds impossible but Italian politicians have turned it before. The betting that Andreotti will succeed, however, is no more than 50-50. He is expected to be at this task for weeks.

If he fails, an election is the ultimate solution. But that would mean even more months of paralyzed government in a country that must make hard decisions quickly on touch issues of economics and law and order.