The score of The Party was by Henry Mancini, including the song "Nothing to Lose." Mancini, commenting on audience reactions, noted, "That's what I get for writing a nice song for a comedy. Nobody's going to hear a note of it." During a scene later in the film, the band can be heard playing "It Had Better Be Tonight," which was a song Henry Mancini composed for the first Pink Panther film.
Hear a bit of "It Had Better Be Tonight" from The Pink Panther:
Here is Claudine Longet performing a version of "Nothing to Lose":
And here, as Hrundi V. Bakshi (Peter Sellers) attempts to retrieve his white loafer from the fountain, the band plays an instrumental version: