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in music, any change in the original text of a musical work making it suitable for a particular purpose, such as its performance by musical amateurs, its use in teaching, or its performance by a different group of instruments.
In Western Europe, the widespread polyphonic arrangement of the melodies of Gregorian chant was the foundation of all polyphonic music until the 16th century. The arrangement of folk melodies, better known as their harmonization, became very important in the 19th and 20th centuries. Many outstanding composers, among them Haydn, Beethoven, Brahms, Balakirev, Rimsky-Korsakov, Tchaikovsky, and Liadov, wrote arrangements of folk melodies. An arrangement of a polyphonic composition for a different instrumentation is often called an adaptation or transposition; when a work is arranged for orchestra it is known as an orchestration. The name “transcription” is often given to an arrangement of a composition, usually written for a single instrument such as piano or violin, that intensifies its virtuoso role while retaining the original instrumentation.
An arrangement of n different things taken m at a time is an ordered set consisting of m elements of the set of n things. The number of such arrangements is equal to
If elements are allowed to be repeated, we speak of an arrangement of n things taken m at a time with repetitions. In this case, the number of arrangements is equal to nm.