8
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From the first Soyuz to the launch vehicles used today, how many launches have been deemed successful?

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9
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As of Sep 24 2013:

The TsSKB Progress site gives the official numbers for all of the R-7 family.

Now, counting only rockets designated Soyuz, that is:

Soyuz-ST, Soyuz-2, Soyuz-U, Soyuz-FG, Soyuz-U2, Soyuz-M, Soyuz-L, Soyuz.

And not counting Vostok, Molnia, Voskhod, Sputnik and R7 MBR, we have

total: 5+23+778+44+70+8+3+32 = 963

failure: 1+21+2 = 24

success: 963 - 24 = 939

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