Problem 39 - Basic Statistics for Business and Economics with Student

Question

USA Today/Gallup Polls developed 15 questions designed to rate the performance of the president of the United States. They will select 10 of these questions. How many different arrangements are there for the order of the 10 selected questions?

Step-by-Step Solution

Answer

There are 3,603,600 different arrangements for the 10 selected questions.

Step 1: Identify the Total and Selected Questions

The total number of questions available is 15, and we want to select and arrange 10 of these questions. This tells us we are dealing with permutations because the order in which we arrange the questions matters.

Step 2: Use the Permutation Formula

To find the number of ways to arrange 10 questions out of 15, we use the permutation formula: \[ P(n, r) = \frac{n!}{(n-r)!} \]where \( n \) is the total number of items (15 in this case) and \( r \) is the number of items to arrange (10 here).

Step 3: Insert the Values into the Formula

Substitute \( n = 15 \) and \( r = 10 \) into the permutation formula:\[ P(15, 10) = \frac{15!}{(15 - 10)!} = \frac{15!}{5!} \]

Step 4: Calculate the Factorials

Calculate the factorials:- \( 15! = 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5! \)- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

Step 5: Solve the Permutation Calculation

Now calculate:\[ \frac{15!}{5!} = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6}{1} = 3,603,600 \]Cancel out the \( 5! \) from the numerator since it is in both the numerator and the denominator.

Key Concepts

Factorials

Factorials are a mathematical concept used to describe the product of all positive integers up to a certain number.
The factorial of a number is denoted with an exclamation mark. For instance, the factorial of 5 is written as 5!.
This means:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

In general, the factorial of a number n is expressed as:

• n! = n × (n-1) × (n-2) × ... × 2 × 1

Factorials grow very rapidly. For example, 10! is 3,628,800.
Another interesting point is that the smallest factorial, by definition, is 0!, which equals to 1.
Factorials are widely used in permutations, combinations, and other areas of combinatorial mathematics, making them an essential tool when arranging or selecting items where order matters.

Permutation Formula

When we talk about permutations, we are focusing on arrangements where the order is crucial.
The permutation formula helps calculate the number of possible ways to arrange a subset of items from a larger set, considering the order of selection.
The formula for permutations is:\[ P(n, r) = \frac{n!}{(n-r)!} \]Here:

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

This formula provides how many different ways you can order \( r \) items out of \( n \) total items.
For example, in the problem where you have 15 questions and you want to select and arrange 10, you apply the formula:\[ P(15, 10) = \frac{15!}{5!} \]This computation shows how many ways the questions can be ordered where every question gets a distinct position in the lineup.
Permutations are beneficial in scenarios like seating arrangements, schedule planning, or conducting surveys where questions need a sequence.

Combinatorial Mathematics

Combinatorial mathematics is a branch of mathematics dedicated to counting, arrangement, and analyzing discrete structures.
It is basically the mathematics of counting and arranging objects.
This field provides the tools and principles needed to solve problems about combinations and permutations.

• Permutations focus on the arrangement of objects where order matters.
• Combinations involve selection where order does not matter.

In our problem, the use of permutations shows a classic combinatorial problem where we determine the number of ways to organize a subset of a larger set.
Combinatorial tools, such as factorials and permutation formulas, help in solving everyday applications. You see these concepts being applied in probability theory, algorithm design, network theory, and even solving puzzles.
Thus, understanding combinatorial mathematics is vital for tackling a wide range of practical and theoretical problems.