4. Probability

Counting

4. Probability

Counting

2:34 minutes

Problem 4.4.23a

Textbook Question

Corporate Officers and Committees The Self Driving Unicycle Company was recently successfully funded via Kickstarter and must now appoint a president, chief executive officer (CEO), chief operating officer (COO), and chief financial officer (CFO), and chief human resources officer (CHR). It must also appoint a strategic planning committee with five different members. There are 15 qualified candidates, and officers can also serve on the committee.

a. How many different ways can the five officers be appointed?

Verified step by step guidance

Step 1: Recognize that the problem involves arranging 5 officers (president, CEO, COO, CFO, and CHR) from a pool of 15 candidates. Since the roles are distinct, the order in which the officers are chosen matters. This is a permutation problem.

Step 2: Use the formula for permutations, which is given by:

P(n, r) = \frac{n!}{(n - r)!}

, where 'n' is the total number of candidates (15) and 'r' is the number of positions to fill (5).

Step 3: Substitute the values into the formula:

P(15, 5) = \frac{15!}{(15 - 5)!} = \frac{15!}{10!}

. This simplifies the calculation by canceling out the factorial terms from 10! downward.

Step 4: Expand the numerator (15!) up to the first 5 terms, as the rest will cancel with the denominator (10!):

15 \times 14 \times 13 \times 12 \times 11

. This gives the total number of ways to arrange the 5 officers.

Step 5: Conclude that the result of this calculation represents the total number of distinct ways to appoint the 5 officers from the 15 candidates.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Permutations

Permutations refer to the different arrangements of a set of items where the order matters. In this context, the appointment of corporate officers is a permutation problem because each position (president, CEO, COO, CFO, CHR) is distinct and the order of selection affects the outcome. The formula for permutations is n! / (n - r)!, where n is the total number of candidates and r is the number of positions to fill.

Combinations

Combinations are selections of items where the order does not matter. In the context of forming a strategic planning committee from the pool of candidates, combinations are used because the arrangement of committee members is irrelevant. The formula for combinations is n! / [r!(n - r)!], where n is the total number of candidates and r is the number of members to choose.

Factorial

A factorial, denoted as n!, is the product of all positive integers up to n. Factorials are essential in calculating permutations and combinations, as they provide the total number of ways to arrange or select items. For example, 5! equals 120, representing the number of ways to arrange five distinct officers. Understanding factorials is crucial for solving problems involving arrangements and selections.

Watch next

Master Introduction to Permutations with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

Radio Station Call Letters Radio and Television station call letters must begin with either K (for stations west of the Mississippi River) or W (for stations east of the Mississippi River) and must include either two or three additional letters. How many different possibilities are there?

Textbook Question

DNA Nucleotides DNA (deoxyribonucleic acid) is made of nucleotides. Each nucleotide can contain any one of these nitrogenous bases: A (adenine), G (guanine), C (cytosine), T (thymine). If one of those four bases (A, G, C, T) must be selected three times to form a linear triplet, how many different triplets are possible? All four bases can be selected for each of the three components of the triplet.

Textbook Question

Teed Off When four golfers are about to begin a game, they often toss a tee to randomly select the order in which they tee off. What is the probability that they tee off in alphabetical order by last name?

Textbook Question

Phone Numbers Current rules for telephone area codes allow the use of digits 2–9 for the first digit, and 0–9 for the second and third digits, but the last two digits cannot both be 1 (to avoid confusion with area codes such as 911). How many different area codes are possible with these rules? That same rule applies to the exchange numbers, which are the three digits immediately preceding the last four digits of a phone number. Given both of those rules, how many 10-digit phone numbers are possible? Given that these rules apply to the United States and Canada and a few islands, are there enough possible phone numbers? (Assume that the combined population is about 400,000,000.)

Textbook Question

Corporate Officers and Committees The Self Driving Unicycle Company was recently successfully funded via Kickstarter and must now appoint a president, chief executive officer (CEO), chief operating officer (COO), and chief financial officer (CFO), and chief human resources officer (CHR). It must also appoint a strategic planning committee with five different members. There are 15 qualified candidates, and officers can also serve on the committee.

b. How many different ways can a committee of five be appointed?

Textbook Question

ATM You want to obtain cash by using an ATM, but it’s dark and you can’t see your card when you insert it. The card must be inserted with the front side up and the printing configured so that the beginning of your name enters first.

a. What is the probability of selecting a random position and inserting the card with the result that the card is inserted correctly?

Textbook Question

c. How many random selections are required to be absolutely sure that the card works because it is inserted correctly?

Textbook Question

Is the Researcher Cheating? You become suspicious when a genetics researcher “randomly” selects numerous groups of 20 newborn babies and seems to consistently get 10 girls and 10 boys. The researcher claims that it is common to get 10 girls and 10 boys in such cases.

a. If 20 newborn babies are randomly selected, how many different gender sequences are possible?

Show more practices