The Metropolitan Museum of Art in June. Credit Hiroko Masuike/The New York Times

Even while struggling with its finances, the Metropolitan Museum of Art set a record for attendance, attracting 6.7 million visitors during the fiscal year that ended on June 30 — the highest number since the museum began tracking admissions more than 40 years ago, the Met announced on Thursday.

The total includes the museum’s three locations — the Met Fifth Avenue, the Met Cloisters and the new Met Breuer, which presents modern and contemporary programming and opened in March.

This year’s attendance — the fifth in a row exceeding six million — was 400,000 higher than that of the previous year. The Met attributed this to an increase of about 200,000 visitors at its Fifth Avenue flagship and the Met Cloisters combined and to 185,000 people taking in the Met Breuer during its first four months.

There are other factors in play: The increase took place in a year when a lawsuit forced the Met to more clearly explain its pay-what-you-like entrance policy, calling its $25 full-admission price “suggested” instead of “recommended.” (Signs now advise visitors that “The amount you pay is up to you.”)

Tourism is also up in New York City overall; NYC & Company, the city’s tourism organization, projects that 59.7 million people will visit New York for calendar year 2016 compared with 58.3 million in 2015.

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For the past three years, the Met has been open seven days a week.

Forty percent of the Met’s visitors in fiscal year 2016 came from New York City and the tristate area; 41 percent from 190 countries besides the United States.

“Our audiences are local, national and international, reflecting the depth and breadth of the extraordinary works of art in our galleries,” Thomas P. Campbell, the Met’s director and chief executive, said in a statement.

The museum’s most popular shows over the last year included its roof garden commission — “Transitional Object (PsychoBarn)” by Cornelia Parker — which has drawn more than 340,000 visitors since it opened on April 19; and the Costume Institute’s exhibition “Manus x Machina: Fashion in an Age of Technology,” which has attracted more than 540,000 visitors since May 5. The fiscal year also included enormous crowds for “China: Through the Looking Glass,” which drew about 815,000 visitors before closing in September.

Of the exhibitions that opened at the Met Fifth Avenue, 21 received more than 100,000 visitors each, including “Sargent: Portraits of Artists and Friends” (254,750); “Artistic Furniture of the Gilded Age” (210,903) and “Ancient Egypt Transformed: The Middle Kingdom” (187,030).

More than half of the exhibitions that received more than 100,000 visitors were made up of artworks from the Met’s own collection, among them: “Masterpieces of Chinese Painting From the Metropolitan Collection” (247,833 visitors to date) and “Grand Illusions: Staged Photography From the Met Collection” (147,313). In the future, the Met may increasingly explore its own collections given the institution’s decision to pull back on the number of shows: loans of artwork are usually a big part of the cost. The Met expects to reduce the number of annual exhibitions over the next few years to about 40 from 55.

The Met said recently that its deficit of about $10 million — which necessitated a recent round of buyouts and layoffs — was the result of several factors, including an increase in younger visitors who pay less at the door.

A decline of “30 to 40 cents per person is material,” Daniel H. Weiss, the Met’s president and chief operating officer, said in April.

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