Everytime we have been there (mostly on weekends) it was free. There is a donation box too.
Pay-what-you-wish General Admission is available at ticket counters inside the Museum, and gives access to 40+ permanent galleries, including the Rose Center for Earth and Space. For admission to special exhibitions, films, and the Space Show, a +One or +All ticket is required. These can be purchased in advance on our website, or in person at the Museum. https://ticketing.amnh.org/
It was free for me because it was a school trip
You will have to buy ticket online which can be standard ticket of €23. Entry with other shows (cost more) is also available. You can also buy tickets from the main reception on the day. Hope this helps.
I think it depends which day you go and if you're American or not. We travelled from the UK and paid.
When were there on Tuesday its was free
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