Problem 53 - Algebra and Trigonometry

Question

These problems involve combinations. Violin Recital A violinist has practiced 12 pieces. In how many ways can he choose eight of these pieces for a recital?

Step-by-Step Solution

Answer

495 ways.

Step 1: Understand the Problem

We need to determine the number of ways to choose 8 pieces from a total of 12 pieces. This is a typical "combinations" problem where order does not matter.

Step 2: Identify the Formula

The formula for combinations is given by: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

Step 3: Assign Values

In this problem, we have \( n = 12 \) and \( r = 8 \). We substitute these values into the formula.

Step 4: Calculate the Factorials

Calculate the factorials for \( n \), \( r \), and \( n - r \):- \( 12! = 479001600 \)- \( 8! = 40320 \)- \( (12-8)! = 4! = 24 \)

Step 5: Apply the Formula

Substitute the factorials into the combination formula:\[ C(12, 8) = \frac{12!}{8! \times 4!} = \frac{479001600}{40320 \times 24} \]

Step 6: Perform the Division

Calculate the exact value of the division:\[ C(12, 8) = \frac{479001600}{967680} = 495 \]

Step 7: Write Down the Answer

The violinist can choose 8 pieces from 12 in 495 different ways.

Key Concepts

Factorials

Factorials are a concept from mathematics that fundamentally deal with multiplication.
They arise frequently in problems involving counting and permutations.
A factorial is the product of an integer and all the integers below it, down to 1.
For example, the factorial of 5 is denoted as 5! and is calculated as:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

The factorial function grows very quickly with larger numbers.
It is essential in various mathematical computations, particularly in scenarios where order is involved.
In the context of combinations, factorials help simplify the counting of possible choices.
The formula for a factorial is given simply as: \[ n! = n imes (n-1) imes (n-2) imes ext{...} imes 1 \] This is a crucial tool in calculating permutations and binomial coefficients.
Understanding how to compute factorials is a building block for mastering more complex problems.

Permutations

Permutations relate to the arrangement of objects where the order matters.
It answers the question, in how many ways can you arrange items in a specific sequence.
For example, consider the letters A, B, and C.
The permutations of these letters taken all at once are ABC, ACB, BAC, BCA, CAB, and CBA.

• These are six different permutations because the order is significant.

Mathematically, the formula for permutations of r items from a set of n is:\[ P(n, r) = \frac{n!}{(n-r)!} \] This formula considers the fact that once you start arranging r items, you have fewer options left with each selection.
Permutations are different from combinations, where order does not matter.
Always remember: more arrangements are possible when the order is important.

Binomial Coefficient

The binomial coefficient is a central concept in combinatorics.
It represents the number of ways to choose a subset of items from a larger set, without regard to the order of selection.
This is often referred to as "n choose r" and is denoted by \( C(n, r) \), or sometimes as \( \binom{n}{r} \).

• The formula is: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]

In this formula:

• \( n \) is the total number of items,
• \( r \) is the number of items to choose,
• \( n! \) stands for the factorial of n, which includes every integer from 1 up to n.

The binomial coefficient is used in polynomial expansions, such as in Pascal's Triangle.
It is a fundamental calculation for determining probabilities in statistics and algorithms.
Understanding this concept allows for solving a multitude of complex counting problems easily.