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Sarfaraz Ahmed's team had to smash the world record of victory victory in a one-day international, but after winning the throw at the Lord, they could not muster enough runs to give them a chance.
Their total of 315 for nine meant that Bangladesh needed only eight runs to finish Pakistan's hopes of progress in the tournament, and they reached it in the other.
If they win on Friday, 1992, Masters Pakistan will end up with the same number of points and win as New Zealand, but go home due to ongoing slips.
New Zealand participates in Australia, India and England next week's semi-finals.
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Pakistan, backed by thousands of passionate flag-waving fans, began curiously slowly in London's sunshine, crawling to 38-1 after 10 overs.
Imam-ul-Haq and Babar Azam eventually went to the gas and reached their 150th partnership for the second 146-ball wicket.
But Babar was out shortly after an impressive 96 out of 98 deliveries.
Babar was beautifully beautiful and hit 11 fours when he and Imam laid the ground for a late-innings blossom.
However, Imam was in another big blow for Pakistan when he believed in his stumps after scoring exactly 100, to leave them 246-3 in the 42nd, with Mohammad Hafeez next time.
A late flower from Imad Wasim, scoring 43 runs from 26 balls, increased the score, but Pakistan lacked the muscle to take the fight away from Bangladesh.
"It feels good when you do something good for the team, but we must win the fight, that's the most important thing," Imam said.
Imam admitted that Pakistan took a realistic approach to the fight and did not aim to 400.
"We must be realistic and consider the conditions," he said. "They are a good side and they bowed tight lines – it was our plan to look for 300 or 320."
Mustafizur Rahman took five wickets to Bangladesh, including one amazingly captured and bowled to dismiss Shadab Khan.
Bangladesh lost Soumya Sarkar for 22 early in their response and Tamim Iqbal was the next to go, leaving them 48-2 in the 11th over.
Pakistan was shot off a bad 105 in their first game, with the Caribbean winning in just 13.4 overs, meaning they took a big hit on the round.
They only won one of their first five games that came back strongly to beat South Africa, New Zealand and Afghanistan to sit fifth in the 10-team table, but eventually they left too much to do.