(Abbreviated) We have used precise radial velocity measurements of subdwarf-B stars from the Palomar-Green catalogue to look for binary extreme horizontal branch (EHB) stars. We identify 36 EHB stars in our sample and find that at least 21 of these stars are binaries. All but one or two of these are new identifications. The minimum binary fraction for EHB stars implied by our survey is 60+-8%. Our survey is sensitive to binaries with orbital periods P less than about 10d. For reasonable assumptions concerning the period distribution and the mass ratio distribution of the binaries, we find that the mean detection efficiency of our survey over this range of orbital periods is 87%. Allowing for this estimated detection efficiency, the fraction of EHB stars which are short-period binaries ($0.03 < P <10d, approximately) is 69+-9%. The value is not strongly dependent on the period distribution below P=10d or the mean companion mass for these short-period binaries. The orbital separation of the stars in these binaries is much less than the size of the red giant from which the EHB star has formed. This is strong evidence that binary star evolution is fundamental to the formation of the majority of EHB stars. If there are also binary EHB stars whose orbital periods are more than about 10d, the fraction of EHB stars whose evolution has been affected by the presence of a companion may be much higher.
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