Standard(s):
[MA2019] FM-19 (9-12) 7 : 7. Develop and use the Fundamental Counting Principle for counting independent and dependent events.
a. Use various counting models (including tree diagrams and lists) to identify the distinguishing factors of a context in which the Fundamental Counting Principle can be applied.
Example: Apply the Fundamental Counting Principle in a context that can be represented by a tree diagram in which there are the same number of branches from each node at each level of the tree.
[MA2019] FM-19 (9-12) 8 : 8. Using application-based problems, develop formulas for permutations, combinations, and combinations with repetition and compare student-derived formulas to standard representations of the formulas.
Example: If there are r objects chosen from n objects, then the number of permutations can be found by the product [n(n-1) ... (n-r)(n-r+1)] as compared to the standard formula n!/(n-r)!
a. Identify differences between applications of combinations and permutations.
b. Using application-based problems, calculate the number of permutations of a set with n elements. Calculate the number of permutations of r elements taken from a set of n elements.
c. Using application-based problems, calculate the number of subsets of size r that can be chosen from a set of n elements, explaining this number as the number of combinations "n choose r."
d. Using application-based problems, calculate the number of combinations with repetitions of r elements from a set of n elements as "(n + r - 1) choose r."
[MA2019] FM-19 (9-12) 9 : 9. Use various counting techniques to determine probabilities of events.