"The only way to be totally free is through education" - Jose Marti

Study Journal Notes

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Lesson 1:

• The Fundamental Counting Principal (FCP) tells us that we can "count" the number of ways something happens by multiply the number of ways the first item can occur by the number of ways the second item can occur and so on

• When solving a problem with digits, know that there are 10 digits that fill up a single space (you need to include the number 0) , but remember that if you are counting the first digit of a number you cannot include 0 in the count.

• ALWAYS start by counting your restrictions, then count the non-restricted items from what is left

Lesson 2:

• Factorial notation is a shorthand way to write a non-repeating count where the items match the number of ways available (for example, 8 ways to organize 8 letters with no repetition).

  • know that 0! = 1​

• Permutation is a shorthand way to write the multiplication of the FCP - use permutation when order matters

  • this is helpful when the number of items does not match the number of ways available (for example, how many ways can 10 letters be arranged in 5 spaces with no repetition)​

Lesson 3:

• When there are restriction on the arrangement of objects ALWAYS deal with the restrictions first, and use the FCP.

  • if the restriction requires for a selection of items to be grouped, the number of ways that the group itself can be organized must be counted, and the number of ways that the remaining items (including the group as a single item) can be arranged must be counted, then the two values will be multiplied together​

    • if the location of the group is specified, then you only need to multiply the number of ways the group is counted by the number of ways the remaining items are counted​

• If you have a selection of items that repeat, set up a fraction where the numerator represents the total available options (use the FCP to find this) and the denominator is the factorial of the number of repetitions for each item

Lesson 4:

• A combination is a selection of items from a group in which order does not matter (we don't worry about re-arranging in unique ways, but we do worry about starting with a unique set of items)

• Combinations are calculated using the permutation formula divided by the factorial of the available spaces

• When restrictions exist, the combination of the restrictions are multiplied by the "leftovers", or the unrestricted remaining items and spaces

• Ensure that the sum of the items in each combination matches the total number of items, and the sum of the spaces in each combination matches the total number of spaces available

• When you must solve for the total number of items, you need to use the combination formula and simplify for "n" --- remember that "n" has to be a whole number!

Lesson 5:

• When you have to solve a combination with the restrictions "at least" or "at most" you need to look at each case that can be included in the restriction and add the combination representing each case together

  • if the restriction given is "at least 1" you can use the difference of combinations to solve (no restrictions minus no items from the "at least" set)​

• In the case of a handshaking problem, there are two "spaces" for the combination - one for each hand involved in a single handshake

Lesson 6:

• Pascal's triangle can be used to determine the coefficients in a binomial expansion.

• The original exponent can be used to determine the reference row and/or the number of terms in the expansion (the row and terms are equal to the exponent plus one)

• The sum of the number in the nth row of Pascal's triangle are equal to 2 to the power of n

• In order to write out the terms of a given binomial expansion, use the combination (starting with "n" choose 0), descending powers of the "x" term (the first term in the binomial) and ascending powers of the "y" term (the second term in the binomial)

• Given the degree of a term, you can find the identity of the term using the general term formula (see lesson 6 notes), and matching the exponents of the variable to the degree of the term

Lesson 7:

• Know that you can use the general term formula to find any particular term of a given binomial expansion

• A constant term is a term with an exponent of zero on every variable

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