score (n.) Look up score at Dictionary.com
late Old English scoru "twenty," from Old Norse skor "mark, tally," also, in Icelandic, "twenty," from Proto-Germanic *skura-, from PIE root *(s)ker- "to cut" (cf. Old English sceran; see shear).

The connecting notion is perhaps counting large numbers (of sheep, etc.) with a notch in a stick for each 20. This counting notion is the origin of the modern sense in sports (1742, originally in whist). In Old French, "twenty" (vint) or a multiple of it could be used as a base, e.g. vint et doze ("32"), dous vinz et diz ("50"). Meaning "printed piece of music" first recorded 1701, from the practice of connecting related staves by scores of lines.
score (v.) Look up score at Dictionary.com
"to cut with incisions or notches," c.1400, see score (n.); the slang sense "achieve intercourse" first recorded 1960. Related: Scored; scoring.