Question

Cory needs to have an equal number of symphonies from Mozart, Beethoven, and Schubert. If he is setting up a schedule of the 12 symphonies to be played, and he has 41 Mozart, 9 Beethoven, and 8 Schubert symphonies from which to choose, how many different schedules are possible? Express your answer in scientific notation rounding to the hundredths place.

Answer 1

4.28×10¹⁷

Explanation:

You want the number of schedules that are possible for 12 symphonies if 4 each are by Mozart, Beethoven, and Schubert, where there are 41 choices by Mozart, 9 by Beethoven, and 8 by Schubert.

Music selection

The number of possible choices of Mozart symphonies is 41C4. Similarly, the numbers of choices by the other composers are 9C4 and 8C4.

Schedule

Any of these sets of choices can be ordered 12! ways. This makes the total number of possible schedules be ...

12! × 41C4 × 9C4 × 8C4 ≈ 4.28×10¹⁷

About 4.28×10¹⁷ different schedules are possible.

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Additional comment

The notation nCk represents the number of ways k items can be chosen from a pool of n items. Its value is ...

nCk = n!/(k!(n-k)!)

We assume that a schedule is different if the same symphonies are in a different order.

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Answer 2

6.12 x 10^11

Explanation:

To find the number of different schedules Cory can create with an equal number of symphonies from Mozart, Beethoven, and Schubert, we can use combinations.

Cory needs to select 4 symphonies from each composer (Mozart, Beethoven, and Schubert) to create a schedule of 12 symphonies in total.

First, let's calculate the number of ways to choose 4 symphonies from each composer:

For Mozart:

41 choose 4 = C(41, 4)

For Beethoven:

9 choose 4 = C(9, 4)

For Schubert:

8 choose 4 = C(8, 4)

Now, we can calculate the total number of different schedules by multiplying these three combinations together because the choices for each composer are independent:

Total number of schedules = C(41, 4) * C(9, 4) * C(8, 4)

Calculating these combinations:

C(41, 4) ≈ 68,109,270

C(9, 4) ≈ 1,386

C(8, 4) ≈ 70

Now, multiply these values together:

Total number of schedules ≈ 68,109,270 * 1,386 * 70 ≈ 6.12 x 10^11 (rounded to the nearest hundredth in scientific notation)

So, there are approximately 6.12 x 10^11 different schedules possible.