LONDON, Oct 25 (Reuters) - Rishi Sunak said on Tuesday he was not daunted by the scale of the challenge as he became Britain's third prime minister in two months, pledging to restore trust, rebuild confidence and lead the country through an economic crisis.

The 42-year-old former hedge fund boss, who has only been in elected politics for seven years, has been tasked with bringing an end to the infighting and feuding at Westminster that has horrified investors and alarmed international allies.

Rishi Sunak said on Tuesday he was not daunted by the scale of the challenge as he became Britain's third prime minister in two months, pledging to restore trust, rebuild confidence and lead the country through an economic crisis.

"I fully appreciate how hard things are," he said outside the prime minister's residence at Downing Street where he shunned the normal tradition of standing with his family and political supporters.

"I understand too that I have work to do to restore trust, after all that has happened. All I can say is that I am not daunted. I know the high office I have accepted and I hope to live up to its demands."

Sunak, one of the richest men in parliament, is expected to slash spending to plug an estimated 40 billion pound ($45 billion) hole in the public finances created by an economic slowdown, higher borrowing costs and a six-month programme of support for people's energy bills.

With his party's popularity in freefall, Sunak will also face growing calls for an election if he moves too far from the policy manifesto that elected the Conservative Party in 2019, when then leader Boris Johnson pledged to invest heavily in the country.

Economists and investors have said Sunak's appointment will calm markets, but they warn that he has few easy options when millions are battling a cost of living crunch.

"I will place economic stability and confidence at the heart of this government's agenda," he said, shortly after he accepted King Charles's request to form a government. "This will mean difficult decisions to come."

He also vowed to put the public's need above politics, reflecting the mounting anger at the sense of perma-crisis that has gripped Britain since the historic 2016 vote to leave the European Union unleashed a battle for the future of the country.

Sunak has warned his colleagues they face an "existential crisis" if they do not help to steer the country through the surging inflation and record energy bills that are forcing many households and businesses to cut back spending.

FINANCIAL TURMOIL

Britain's youngest prime minister for more than 200 years and its first leader of colour, he replaced Liz Truss who resigned after 44 days following a "mini budget" that sparked turmoil in financial markets.

He will now need to review all spending, including on politically sensitive areas such as health, education, defence, welfare and pensions.

Reflecting the near constant state of turmoil in British politics this year, politicians, journalists and photographers once again crammed into Downing Street on Tuesday to hear a departing speech from Truss, and an arrival one from Sunak.

Truss was applauded by politicians as she arrived outside her office and residence to speak, seven weeks to the day after she took office.

She did not apologise for her short tenure despite it sparking a collapse in the pound and a surge in borrowing costs and mortgage rates.

Sunak paid tribute to Truss and said her plan to reignite economic growth had not been wrong, but he said mistakes were made. "And I have been elected as leader of my party and your prime minister, in part to fix them," he said.

Sunak will now start forming his cabinet, with some Conservative lawmakers hoping he will include politicians from all wings of the party.

He is expected to retain Jeremy Hunt as finance minister after the former foreign and health secretary helped calm volatile bond markets by ripping up most of Truss's economic programme.

Investors will also want to know if Sunak still plans to publish a new budget alongside borrowing and growth forecasts on Oct. 31, which would help inform the Bank of England's interest rate decision on Nov. 3.

"Adults have taken charge again," Michael O'Leary, the boss of airline Ryanair, said.

POLITICAL MACHINATIONS

Sunak, a Goldman Sachs analyst who only entered parliament in 2015, must unite his party, aware that voters are increasingly angry over the antics at Westminster as the economy heads for recession.

He was blamed by many in the party when he quit as finance minister in the summer, triggering a wider rebellion that brought down Johnson.

While many expressed relief that the party settled on a new leader quickly, a sense of distrust remains among some while others questioned whether struggling families would relate, or ever vote, for a multimillionaire.

"I think this decision sinks us as a party for the next election," one Conservative lawmaker told Reuters.

Historian and political biographer Anthony Seldon told Reuters that Sunak would also be constrained by the mistakes of his predecessor.

"There is no leeway on him being anything other than extraordinarily conservative and cautious," he said.

Many politicians and officials abroad, having watched as a country once seen as a pillar of economic and political stability descended into brutal infighting, welcomed Sunak's appointment.

Sunak, a Hindu, also becomes Britain's first prime minister of Indian origin.

U.S. President Joe Biden described it as a "groundbreaking milestone", while leaders from India and elsewhere welcomed the news. Sunak's billionaire father-in-law, N.R. Narayana Murthy, said he would serve the United Kingdom well.

"We are proud of him and we wish him success," the founder of software giant Infosys said in a statement.

($1 = 0.8864 pounds)

Reuters