Processes in Microsoft 365 for setting up Office apps, redeeming product keys, and activating licenses.
The Home and Business 2010 license allows you to have Office active on up to 2 computers at the same time. You are allowed to "transfer" the licesne from one of the 2 machines by making Office unusable on that computer (typically that is either uninstall Office 2010, or the computer is "dead"). Then you can install and activate on the new computer.
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BUT ...
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That is how is used to work before Office 2010 reached end of life in Oct 2020. Since then, many people have been reporting problems trying to activate Office 2010 on new computers. In the past, MS allowed one or two "grace" installations. But now it appears they have reduced that grace to 0. Which means you would have to contact MS Activation support, by doing a phone activation, to get them to reset your product key to allow the activation on the new computer to work. But, now 2010 is no longer supported, so Activation support refuses to help. You are out of luck.
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Bottom line, you can try to transfer one of your activation count to the new machine. If it works, great, if not you are SOL.
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Sorry for the bad news. MS is being a PITA, doing everything it can to "encourage" people to spend more money on Office. From MS's point of view, preferably on 365, but they are more than happy to take your money for onetime Payment Office 2019.
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If you do need to buy a new "consumer" Office license here are a couple of points to consider.
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Ask around your "family" (including "extended family" and friends) to find out if anyone has a 365 Family subscription with an unused "share" they can grant you. Or find someone with 365 Personal who would be willing to share the cost of upgrading from Personal to Family. 2 people sharing the cost of Family is cheaper than both paying for Personal.
The other is, if you decide that onetime payment 2019 is good enough (installs on only one computer, unlike 2010 and earlier), consider using a monthly subscription of 365 Personal for a short time. MS has quietly announced that Office "V-Next" (2022?) is going to be released "sometime this year". Educated guess that would be July or October. By paying for 365 monthly (or even just 1 year) until V-Next is released, you can then buy the new one-time payment version and get an extra 3 years after 2019 reaches it's end-of-life in October 2025, plus some current 365-only features that are not in 2019.