Problem 34 - Basic Statistics for Business and Economics with Student

Question

Solve the following: a. 20!/17! b. \({ }_{9} P_{3}\) c. \({ }_{7} C_{2}\)

Step-by-Step Solution

Answer

a. 6840; b. 504; c. 21.

Step 1: Understanding Factorials

To solve part (a) of the exercise, recall that the factorial of a number \( n \) is the product of all positive integers less than or equal to \( n \). Thus, \( 20! = 20 \times 19 \times \ldots \times 1 \) and \( 17! = 17 \times 16 \times \ldots \times 1 \). Since \( 17! \) is a common factor in both numerator and denominator, \( \frac{20!}{17!} = 20 \times 19 \times 18 \).

Step 2: Calculating 20!/17!

Compute \( 20 \times 19 \times 18 \). This simplifies to:\[ 20 \times 19 = 380 \] \[ 380 \times 18 = 6840 \] Thus, \( \frac{20!}{17!} = 6840 \).

Step 3: Understanding Permutations

For part (b), the notation \( {}_{n}P_{r} \) represents permutations of \( n \) items taken \( r \) at a time and is calculated as \( \frac{n!}{(n-r)!} \). For \( {}_{9}P_{3} \), this means we calculate \( \frac{9!}{6!} \).

Step 4: Calculating 9P3

Using the formula from the previous step, calculate \( \frac{9!}{6!} \): Since \( 9! = 9 \times 8 \times 7 \times 6! \), cancel \( 6! \) to get: \( 9 \times 8 \times 7 = 504 \). Thus, \( {}_{9}P_{3} = 504 \).

Step 5: Understanding Combinations

For part (c), the notation \( {}_{n}C_{r} \) represents combinations of \( n \) items taken \( r \) at a time and is calculated as \( \frac{n!}{r!(n-r)!} \). For \( {}_{7}C_{2} \), this means we calculate \( \frac{7!}{2!5!} \).

Step 6: Calculating 7C2

Calculate \( \frac{7!}{2!5!} \):Since \( 7! = 7 \times 6 \times 5! \), cancel \( 5! \) to get: \( \frac{7 \times 6}{2!} = \frac{42}{2} = 21 \). Thus, \( {}_{7}C_{2} = 21 \).

Key Concepts

Permutations

Permutations are a crucial concept in mathematics and statistics when it comes to arranging a set of items. A permutation is an ordered arrangement of elements. The order of arrangement is significant. Consider the permutation formula, which is denoted as \( {}_{n}P_{r} \). It represents the number of ways to arrange \( n \) items taken \( r \) at a time and is calculated using the formula:\[ {}_{n}P_{r} = \frac{n!}{(n-r)!} \]where \( n! \) (read as "n factorial") is the product of all positive integers up to \( n \). Keep in mind that the factorial function is only defined for non-negative integers.

• Permutations are used when the order matters.
• Each different ordering of the same set of items counts as a distinct permutation.
• Common examples include arranging books on a shelf or the order of runners in a race.

Let's look at an example: For \( {}_{9}P_{3} \), meaning 9 items arranged 3 at a time, we use the formula:\[ \frac{9!}{6!} \]Since \( 9! = 9 \times 8 \times 7 \times 6! \), we can cancel \( 6! \) and the result is \( 9 \times 8 \times 7 = 504 \). Hence, there are 504 ways to arrange 9 items in sets of 3.

Combinations

Combinations refer to the selection of items without regard to the order. The concept of combinations is fundamental when you need to count the number of ways to select items from a larger set. The formula for combinations, represented as \( {}_{n}C_{r} \), is:\[ {}_{n}C_{r} = \frac{n!}{r!(n-r)!} \]Here, \( n! \) is the factorial of \( n \), and \( r! \) is the factorial of \( r \). To better understand, consider why the order doesn't matter in combinations, unlike permutations.

• Combinations are used when the order of selection is irrelevant.
• This could include selecting members for a committee or picking cards from a deck without considering the order.
• Each unique group is only counted once, regardless of order.

For example, find \( {}_{7}C_{2} \) for 7 items taken 2 at a time:\[ \frac{7!}{2!5!} = \frac{7 \times 6 \times 5!}{2! \times 5!} \]Cancel out \( 5! \) to simplify to \( \frac{42}{2} = 21 \). Thus, there are 21 ways to choose two items from seven without regard to order.

Basic Statistics

Basic statistics often involve concepts like permutations and combinations since they form the backbone of probability and data analysis. These concepts help in understanding how data can be arranged or grouped.

• The use of factorials in basic statistics is essential for calculating permutations and combinations.
• Factorials are symbolized by \( n! \) and represent the product of all positive integers up to \( n \).
• In statistics, these calculations help in determining likelihoods and making predictions from data samples.

Consider factorials in calculating different possible arrangements of a dataset, or in determining the likelihood of particular outcomes when handling multiple variables. Statistics also uses these calculations in hypothesis testing and probability distributions, often requiring an understanding of how many ways certain outcomes can occur based on a dataset's size and sample selections.