Problem 16 - Calculus for Biology and Medicine

Question

You have just enough time to play 4 songs out of 10 from your favorite CD. In how many ways can you program your CD player to play the 4 songs?

Step-by-Step Solution

Answer

There are 5040 ways to program the CD player to play 4 songs from 10.

Step 1: Understanding the problem

The task is to determine the number of ways to select 4 songs from a total of 10 songs on a CD, considering that the order of songs matters.

Step 2: Applying permutation formula

Since the order in which the songs are played matters, we use the permutation formula for selecting 4 songs from 10. The formula for permutation is given by \[ P(n, r) = \frac{n!}{(n-r)!} \], where \( n \) is the total number of items to choose from and \( r \) is the number of items to choose.

Step 3: Substituting values

Set \( n = 10 \) and \( r = 4 \) in the permutation formula: \[ P(10, 4) = \frac{10!}{(10-4)!} = \frac{10!}{6!} \].

Step 4: Calculating factorial values

First, calculate the factorials: 10! = 10 × 9 × 8 × 7 × 6! 6! = 720 (which cancels out in the fraction), thus we only need to compute 10 × 9 × 8 × 7.

Step 5: Final calculation

Multiply the remaining values after cancelling 6!: 10 × 9 × 8 × 7 = 5040.

Key Concepts

Factorial

Factorial is a fundamental concept in mathematics, especially in permutations and combinations. It is denoted by an exclamation mark \(!\). When you see \( n! \), it means you are multiplying a series of descending natural numbers starting from \( n \). For instance, \( 5! \) is the same as multiplying 5 \( \times \) 4 \( \times \) 3 \( \times \) 2 \( \times \) 1, equaling 120.

Factorials are crucial when dealing with arrangements since they help in finding how many ways you can organize a set of items. With larger numbers, factorial calculations can become hefty, but tools like calculators often have a factorial function to do the work swiftly.

Permutation Formula

The permutation formula is a powerful tool to determine how many ways you can arrange a smaller subset of items from a larger set, especially when the order matters.

It is expressed as \[P(n, r) = \frac{n!}{(n-r)!}\]

• \( n \) is the total number of items.
• \( r \) is the number of items to arrange.

As illustrated in the exercise, with 10 songs and a requirement to choose 4, the formula calculates the permutations as follows: \[ \frac{10!}{(10-4)!} \], which simplifies the arrangement to just computing the product of the successive 4 numbers, from 10 down to 7. This makes it especially useful for problems involving sequences such as phone numbers, passwords, or in this case, song arrangements on a CD.

Combinatorics

Combinatorics is the broad study of counting, arrangement, and combination of elements within certain constraints. It’s a key area of mathematics used to solve problems involving selections and orderings.

In the example problem of selecting songs, it’s a matter of choosing and ordering. Combinatorics gives us a structured way to understand these scenarios through formulas and logical reasoning. Within combinatorics, permutations are used when order matters, while combinations are used when order doesn't matter.

The problem of programming the CD to play 4 specific songs from a list of 10 is a classic example of a permutation problem since the sequence in which the songs are played is important to the outcome. Therefore, understanding combinatorics not only helps in mathematical problems but also in everyday situations like organizing schedules or planning activities efficiently.