If one event can occur in [latex]m[\/latex] ways and a second event with no common outcomes can occur in [latex]n[\/latex] ways, then the first or second event can occur in [latex]m+n[\/latex] ways.<\/li>\n \t
If one event can occur in [latex]m[\/latex] ways and a second event can occur in [latex]n[\/latex] ways after the first event has occurred, then the two events can occur in [latex]m\\times n[\/latex] ways.<\/li>\n \t
A permutation is an ordering of [latex]n[\/latex] objects.<\/li>\n \t
If we have a set of [latex]n[\/latex] objects and we want to choose [latex]r[\/latex] objects from the set in order, we write [latex]P\\left(n,r\\right)[\/latex].<\/li>\n \t
Permutation problems can be solved using the Multiplication Principle or the formula for [latex]P\\left(n,r\\right)[\/latex].<\/li>\n \t
A selection of objects where the order does not matter is a combination.<\/li>\n \t
Given [latex]n[\/latex] distinct objects, the number of ways to select [latex]r[\/latex] objects from the set is [latex]\\text{C}\\left(n,r\\right)[\/latex] and can be found using a formula.<\/li>\n \t
A set containing [latex]n[\/latex] distinct objects has [latex]{2}^{n}[\/latex] subsets.<\/li>\n \t
For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations.<\/li>\n \t
[latex]\\left(\\begin{gathered}n\\\\ r\\end{gathered}\\right)[\/latex] is called a binomial coefficient and is equal to [latex]C\\left(n,r\\right)[\/latex].<\/li>\n \t
The Binomial Theorem allows us to expand binomials without multiplying.<\/li>\n \t
We can find a given term of a binomial expansion without fully expanding the binomial.<\/li>\n<\/ul>\n
Glossary<\/h2>\nAddition Principle<\/strong> if one event can occur in [latex]m[\/latex] ways and a second event with no common outcomes can occur in [latex]n[\/latex] ways, then the first or second event can occur in [latex]m+n[\/latex] ways\n\nbinomial coefficient<\/strong> the number of ways to choose r<\/em> objects from n<\/em> objects where order does not matter; equivalent to [latex]C\\left(n,r\\right)[\/latex], denoted [latex]\\left(\\begin{gathered}n\\\\ r\\end{gathered}\\right)[\/latex]\n\nbinomial expansion<\/strong> the result of expanding [latex]{\\left(x+y\\right)}^{n}[\/latex] by multiplying\n\nBinomial Theorem<\/strong> a formula that can be used to expand any binomial\n\ncombination<\/strong> a selection of objects in which order does not matter\n\nFundamental Counting Principle<\/strong> if one event can occur in [latex]m[\/latex] ways and a second event can occur in [latex]n[\/latex] ways after the first event has occurred, then the two events can occur in [latex]m\\times n[\/latex] ways; also known as the Multiplication Principle\n\nMultiplication Principle<\/strong> if one event can occur in [latex]m[\/latex] ways and a second event can occur in [latex]n[\/latex] ways after the first event has occurred, then the two events can occur in [latex]m\\times n[\/latex] ways; also known as the Fundamental Counting Principle\n\npermutation<\/strong> a selection of objects in which order matters\n","rendered":"
Key Equations<\/h2>\n
\n\n
\n
number of permutations of [latex]n[\/latex] distinct objects taken [latex]r[\/latex] at a time<\/td>\n
If one event can occur in [latex]m[\/latex] ways and a second event with no common outcomes can occur in [latex]n[\/latex] ways, then the first or second event can occur in [latex]m+n[\/latex] ways.<\/li>\n
If one event can occur in [latex]m[\/latex] ways and a second event can occur in [latex]n[\/latex] ways after the first event has occurred, then the two events can occur in [latex]m\\times n[\/latex] ways.<\/li>\n
A permutation is an ordering of [latex]n[\/latex] objects.<\/li>\n
If we have a set of [latex]n[\/latex] objects and we want to choose [latex]r[\/latex] objects from the set in order, we write [latex]P\\left(n,r\\right)[\/latex].<\/li>\n
Permutation problems can be solved using the Multiplication Principle or the formula for [latex]P\\left(n,r\\right)[\/latex].<\/li>\n
A selection of objects where the order does not matter is a combination.<\/li>\n
Given [latex]n[\/latex] distinct objects, the number of ways to select [latex]r[\/latex] objects from the set is [latex]\\text{C}\\left(n,r\\right)[\/latex] and can be found using a formula.<\/li>\n
A set containing [latex]n[\/latex] distinct objects has [latex]{2}^{n}[\/latex] subsets.<\/li>\n
For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations.<\/li>\n
[latex]\\left(\\begin{gathered}n\\\\ r\\end{gathered}\\right)[\/latex] is called a binomial coefficient and is equal to [latex]C\\left(n,r\\right)[\/latex].<\/li>\n
The Binomial Theorem allows us to expand binomials without multiplying.<\/li>\n
We can find a given term of a binomial expansion without fully expanding the binomial.<\/li>\n<\/ul>\n
Glossary<\/h2>\n
Addition Principle<\/strong> if one event can occur in [latex]m[\/latex] ways and a second event with no common outcomes can occur in [latex]n[\/latex] ways, then the first or second event can occur in [latex]m+n[\/latex] ways<\/p>\n
binomial coefficient<\/strong> the number of ways to choose r<\/em> objects from n<\/em> objects where order does not matter; equivalent to [latex]C\\left(n,r\\right)[\/latex], denoted [latex]\\left(\\begin{gathered}n\\\\ r\\end{gathered}\\right)[\/latex]<\/p>\n
binomial expansion<\/strong> the result of expanding [latex]{\\left(x+y\\right)}^{n}[\/latex] by multiplying<\/p>\n
Binomial Theorem<\/strong> a formula that can be used to expand any binomial<\/p>\n
combination<\/strong> a selection of objects in which order does not matter<\/p>\n
Fundamental Counting Principle<\/strong> if one event can occur in [latex]m[\/latex] ways and a second event can occur in [latex]n[\/latex] ways after the first event has occurred, then the two events can occur in [latex]m\\times n[\/latex] ways; also known as the Multiplication Principle<\/p>\n
Multiplication Principle<\/strong> if one event can occur in [latex]m[\/latex] ways and a second event can occur in [latex]n[\/latex] ways after the first event has occurred, then the two events can occur in [latex]m\\times n[\/latex] ways; also known as the Fundamental Counting Principle<\/p>\n
permutation<\/strong> a selection of objects in which order matters<\/p>\n\n\t\t\t \n\t\t\t