Combinations

swerve wrote: ↑

Fri Feb 26, 2021 10:41 am

A pianist is planning the order in which they play their recital. They will play 6 pieces by 6 different composers: Brahms, Chopin, Haydn, Liszt, and Schubert. The recital will be in two halves with 3 pieces before a short interval and 3 pieces straight after. The pianist decides that the final piece in each half will be by one of Liszt, Brahms, or Mozart. How many different different orders are there for the pianist's recital?

A. 48
B. 72
C. 120
D. 144
E. 720

The OA is D

Solution:

First, note that the 6th composer in the initial list is missing; later information given in the problem allows us to infer that this 6th composer must be Mozart.

The ordering of the musical recital is ___, ___, ___, interval, ___, ___, ___

We see that positions 3 and 6 can be filled by any of 3 composers (Liszt, Brahms, or Mozart), so for just those two positions, we have 3P2 = 6 ways to fill them. For the remaining positions (1, 2, 4, and 5), we have 4! = 24 ways to arrange the four remaining composers. Thus, the total number of arrangements of the selections for the recital is 3P2 x 4! = 6 x 24 = 144.

Answer: D