Count individual entries with 2 criteria

Hello again, Excellent people!

I`m having some problems with counting individual entries in a huge table [120969 rows / 20 columns and counting].
I`m trying to extract individual entries from a specific month (month \ year varies) (and then year), where the entries are tied with another column.
[how many individual vehicles (A:A) does a company [C:C] has for specified month/year [B:B])

Unique formula doesn`t seem to work on my MS 365 ProPlus, so I've tried SUMPRODUCT and other examples from the web.

Formula:

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and

Formula:

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These formulas should work, but I`m doing something wrong apparently.
I have attached a scaled down version of the table. Any help would be greatly appreciated.

Thank you!

Formula seems to be working fine on the scaled down table, but when applying it to the large table it returns 0,00 and all active workbooks function as if on a Pentium 2 that runs on solar power on the dark side of the moon. Guess I`ll have to try something else or scrap this one.
Thanks for the prompt reply!

Me again!
Finally managed to adapt samba_ravi`s formula (Thanks again!) to my table, but it takes 2-5-10 minutes to calculate something and every opened Workbook freezes or stops responding.
I work on a i5 - 7500 3.40Ghz with 16 GB RAM so this shouldn't happen.
Anyone knows what's wrong?

Here is the updated formula, with more criteria, just in case someone else needs it:

Formula:

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I guess this thread is Solved and can be closed, if no other replies come my way.

Thanks!

Yes! I use the "big table" just to extract data to other Sheets.

Thank you for the reply and the formulas, bebo021999!
As in samba_ravi's case, the formulas are working fine, but I am unable to extend them to the table, as it keeps crashing from "lack of memory".
I can`t go MSO 64 bit and 120979 rows x 2 helpers and one additional Sheet full of complex formulas, seems to be a far to difficult task for my i5 and it`s 16 GB of RAM.
I`ll keep trying to adapt what you two have provided and hope for the best.
My thanks!

Hello, kvsrinivasamurthy! Thanks for the reply!
Unfortunately for now I`m confined to a laptop that can`t open the big file, so I can't test your solution.
F3 should be =3, because there are 3 unique entries in A:A for G2, that have dates in B:B after value in F2 (1.01.2020 + 12 months). *ILO had 3 cars from 1.01.2020 until today, but only 2 in October (10.2020). So, E3 should count one B101AAA and one B110AAA whileF3 should add one B102AAA.
Thank you again! Stay safe!