Factorials are just products indicated by an exclamation mark. For example, 4! (read as "four factorial") means 1 x 2 x 3 x 4 = 24.

In general, n! means the product of all the whole numbers from 1 to n, that is to say: n! = 1 x 2 x 3 x ... x n.

Permutation & Combination

A combination is a selection of items from a collection, such that the order of selection does not matter. On the other hand, a permutation is an ordered selection of items. For example, the fruit salad with apples, grapes and bananas is a combination, and the combination of 472 to the safe is a permutation, because the order of the numbers makes a difference.

There are 2 types of combinations, depending on if repetition is allowed (example: pocket coins of 5, 10, 25, 25), or not allowed (example: lottery numbers).

There are also 2 types of permutations, depending on if repetition is allowed (example: numbers in a combination lock), or not allowed (example: the placement of participants in a race).

Formula for calculating permutations with repetition

Where P is the total number of possible choices, n is the number of total items, and r is the number of selected items

Formula for calculating permutations without repetition

Where P is the total number of possible choices, n is the number of total items, and r is the number of selected items.

Formula for calculating combinations without repetition

Where C is the total number of possible choices, n is the number of total items, and r is the number of selected items.

The formula for calculating combinations with repetition is a lot more complicated and will not be covered here.

Probability

In real life, many events can't be predicted with total certainty. The best we can say is how likely they will happen. When we use the words probably, likely, possibly, etc., we are using the concept of probability. Probability is just the chance of some events happening.

How do we calculate the value of a probability? We know that when tossing a coin, the chance of getting heads or tails are the same, which are both 50%, or 0.5.

In general, we calculate the value of the probability of an event happening by dividing the number of ways the event can happen with the total number of outcomes.

For example, suppose we have a dice with numbers 1, 2, 3, 4, 5, 6 on its faces. When we throw the dice, the probability of getting number 6 will be 1 out of 6, or 1/6. The probability of getting 1 or 6 will be (1+1)/6 = 2/6, or 1/3.

In calculating probability, you will find the addition principle and the multiplication principle very useful:

Addition Principle: if one event can occur in m ways and a second event with no common outcomes can occur in n ways, then the first or second event can occur in m + n ways.

As an example: suppose a student is shopping for a smart phone. He is deciding among 4 types of Android phones and 2 types of iPhones (iPhone 11 or iPhone XS). The total number his choices is 4+2 = 6.

Multiplication Principle: if one event can occur in m ways and a second event can occur in n ways after the first event has occurred, then the two events can occur in m times n ways. This is also known as the Fundamental Counting Principle.

As an example: suppose Alice has 3 skirts, 4 blouses, and 2 sweaters for her to choose from. She will need to choose 1 skirt and 1 blouse for each outfit and she will then need to decide whether to wear a sweater (and if yes, which one). We can use the Multiplication Principle to find out the total number of possible outfits.

Principle of Inclusion and Exclusion (PIE)

Another very useful technique in combinatory math and probability questions is the use of the principle of inclusion and exclusion (PIE). PIE is a counting technique that calculates the number of elements that satisfy at least one of several properties while ensuring that elements satisfying more than one property are not double-counted.

For example, suppose we want to know how many integers between 1 and 20 are multiples of 2 or multiples of 3. We know that an integer can be both the multiple of 2 and of 3. Between 1 and 20, these integers are 6, 12, and 18. The integers between 1 and 20 that are the multiple of 2 include 2, 4, 6, 8, 10, 12, 16, 18, and 20; the integers between 1 and 20 that are the multiple of 3 include 3, 6, 9, 12, 15, and 18. So we have 9 + 5 - 3 = 11 integers between 1 and 20 that are multiples of 2 or 3.

Complementary counting

This is another useful technique in counting. According to this technique, instead of counting the number of cases we are asked to count, we can count the number of opposite cases, and then subtract that from the total number of cases.

For example: Out of 200 students, there are 100 taking Calculus, 70 taking algebra, and 30 taking both.  How many students are taking neither?

The opposite of taking neither classes is taking either classes. According to PIE, we know that number is 100+70 - 30 = 140. So according to Complementary Counting, the opposite of that, ie, the number of students taking neither classes will be 200 - 140 = 60.