According to a probabilistic principle sometimes known as the Fundamental or Multiplication Counting Rule
Instruction Details
1. Combinatorics is a subfield of mathematics that focuses on counting, essentially enumerating how many arrangements are possible, including contexts where arrangements are ordered or unordered.
Agree or disagree?
2. According to a probabilistic principle sometimes known as the Fundamental or Multiplication Counting Rule, when an event has m options during a first phase and n options during a second phase, then the total number of options is m x n or m n.
Agree or disagree?
3. The Fundamental Counting Rule principle applies even when there are more than two options. Simply multiply all the options per phase or stage together for a compound event.
Agree or disagree?
4. An ordered counting arrangement is known as a permutation, symbolized as or , where n is the total number, and r is the number being ordered in some special manner.
Agree or disagree?
5. An unordered counting arrangement is known as a combination, symbolized as or , where n is the total number, and r is the number being arranged in some unordered manner.
Agree or disagree?
6. Factorials are a special product of counting numbers from 1 to the number of interest, n. This can be written as n!
Agree or disagree?
7. Factorials are a special product of counting numbers from 1 to the number of interest, n.
The spreadsheet function is =FACT(
Agree or disagree?
8. Attempt the same problem using the permutation spreadsheet function, =PERMUT(12,12)
Since both involve ordered arrangements of 12 customers in 12 spots in line, you do not get the same answer.
Agree or disagree?
9. FALSE or TRUE: Probability is the likelihood or chance an outcome or event will happen; it is typically represented in numerical form as a ratio or fraction, a decimal or percentage, but can also be described verbally (ex: certain or very unlikely).
10. Probability can be expressed basically as a ratio: desired outcomes divided by total outcomes. Consider it as a fraction where the numerator, or top, includes the number of outcomes of interest, and the denominator, or bottom, includes the total number of outcomes.
Seems reasonable? Common sense? Agreed?
a. Yes
b. No
Answers
1. Combinatorics and Counting Arrangements
Agree
Combinatorics is indeed a subfield of mathematics that deals with counting and arranging objects in various ways. It involves both ordered and unordered arrangements, and its principles are applied in solving problems involving permutations, combinations, and other counting scenarios.
2. Multiplication Counting Rule
Agree
The statement is correct. The Multiplication Counting Rule states that if an event has “m” options during a first phase and “n” options during a second phase, then the total number of options for the combined event is the product of “m” and “n,” which is represented as “m x n” or “mn.”
3. Fundamental Counting Rule for Multiple Options
Agree
Yes, the Fundamental Counting Rule applies even when there are more than two options. You can multiply all the options per phase or stage together for a compound event involving multiple phases or stages.
4. Permutations
Agree
An ordered counting arrangement is indeed known as a permutation. The symbol for permutations is “nPr” or “P(n, r),” where “n” is the total number of items, and “r” is the number of items being ordered in a specific manner.
5. Combinations
Agree
An unordered counting arrangement is known as a combination. The symbol for combinations is “nCr” or “C(n, r),” where “n” is the total number of items, and “r” is the number of items being selected in an unordered manner.
6. Factorials
Agree
Factorials are a special product of counting numbers from 1 to the number of interest, “n.” They are indeed written as “n!” and represent the product of all positive integers up to “n.”
7. Factorials Spreadsheet Function
Agree
The statement is correct. The spreadsheet function for calculating factorials is “=FACT(n),” where “n” is the number for which you want to compute the factorial.
8. Permutations Spreadsheet Function
Agree
The statement is accurate. The permutation spreadsheet function “=PERMUT(12, 12)” involves ordered arrangements of 12 customers in 12 spots in line. Since order matters, this calculation will yield a different result from a combination calculation.
9. Probability Definition
Agree
True. Probability represents the likelihood or chance of a particular outcome or event occurring. It can be expressed numerically as a ratio, fraction, decimal, or percentage. Different outcomes’ probabilities sum up to 1, and they can range from “certain” (probability of 1) to “very unlikely” (probability close to 0).
10. Probability Calculation
Agree
Yes, the description of probability as a ratio of desired outcomes to total outcomes is accurate. The probability of an event is calculated by dividing the number of favorable outcomes (numerator) by the total number of possible outcomes (denominator). This representation forms the basis for basic probability calculations.
a. Yes
Complete Answer:
