Question

A concert to raise money for an economics prize is to consist of 6 works: 3 overtures, 2 sonatas, and a piano concerto. (a) In how many ways can the program be arranged? (b) In how many ways can the program be arranged if a sonata must come first? (a)way(s)________ (b)way(s)_________

Answer 1

There are 60 ways to arrange the concert program in total, and 60 ways to arrange it if a sonata must come first.

Step-by-step explanation:

Arranging a Concert Program

To answer these questions, we need to use principles of combinatorics, a field of mathematics that deals with counting combinations and permutations of sets.

(a) Total Arrangements

We have 6 works to arrange: 3 overtures, 2 sonatas, and 1 piano concerto. Since the overtures and sonatas are not distinct from each other, we use the formula for permutations of a multiset:

Total arrangements =
\(\frac{6!}{3! \times 2! \times 1!}\)

Calculating that gives us:

Total arrangements =
\(\frac{720}{12}\) = 60 ways.

(b) Sonata First

if one of the sonatas must come first, we are left with 5 positions to fill. We treat the first sonata as fixed, and arrange the remaining 5 works:

Arrangements with sonata first =
(\frac{5!}{2! \times 1!}\)

Which simplifies to:

Arrangements with sonata first =
\(\frac{120}{2}\) = 60 ways.