Question

The composer Beethoven wrote 9 symphonies, 5 piano concertos (music for piano and orchestra), and 32 piano sonatas (music for solo piano). a. How many ways are there to play first a Beethoven symphony and then a Beethoven piano concerto?b. The manager of a radio station decides that on each successive evening (7 days per week), a Beethoven symphony will be played followed by a Beethoven piano concerto followed by a Beethoven piano sonata. For how many years could this policy be continued before exactly the same program would have to be repeated?

Answer 1

a) 45 ways

b) 3.95 years

Almost 4 years!

Explanation:

The problem of how many ways different selections can be made with order important is a permutation and combination problem

a) The number of ways there are to play first a Beethoven symphony and then a Beethoven piano concerto.

There are 9 Beethoven symphonies to select from.

There are 5 Beethoven piano concertos to select from.

The total number of ways the selection can be made = ⁹C₁ × ⁵C₁ = 45 ways.

b) How many ways to play a Beethoven symphony, followed by a piano concerto and then a piano sonata.

There are 9 Beethoven symphonies to select from.

There are 5 Beethoven piano concertos to select from.

There are 32 Beethoven piano sonatas to select from.

Total number of ways = ⁹C₁ × ⁵C₁ × ³²C₁ = 1440 ways.

With each day, taking each way, this means, it'll take 1440 days to exhaust the different number of ways to do this

There are 7 days in a week,

So, 1440 days has (1440/7) weeks; 205.71 weeks

52 weeks = 1 year

205.71 weeks = (205.71/52) = 3.95 years

Almost 4 years!