Surveying Different Counting Methods

Question 1

Putting the Counting Principle to Work

Answer 1

According to the counting principle, if an event has m** possible outcomes and another independent event has **n possible outcomes, then the total possible outcomes of both events occurring together is mn.

Example:

Suppose ou have five shirts, two pairs of pants, and two jackets. How many different outfits can you put together?

To answer this question, apply the counting principle:

1) Make a space for each item that can change.

In this case, you have three spaces: one for shirts, one for pants, and one for jackets.

2) In each space, jot down the number of options.

You now have something like this:

Shirts Pants Jackets

5 2 2

3) Multiply the numbers

5 x 2 x 2 = 20 different outfits

* Use the same technique to calculate the possible combinations you have when rolling two six-sided dice. Each die has 6 possible outcomes, so the total number of different ways the numbers on the dice can be combined are: 6 x 6 = 36

Question 2

When Order Matters: Permutations

Answer 2

A permutation is a change in the arrangement of a given number of items or events. If the order in which items are arranged or in which events occur matters, you’re looking a a permutation problem.

Whenever a question asks for the possible number of ways a certain number of objects may be arranged or events may occur, you may use the following formula:

(n)(n-1)(n-2)(n-3)…

In which n represents the total number of items or events.

Mathematics uses the factorial as a convenient way to represent the product of integers up to and including a specific integer, so 3!, which stands for “factorial,” is 3 x 2 x 1 = 6.

Factorials play a key role in calculating possible permutations and combinations.

0! breaks the rule. 0! = 1

You can solve simple permutation problems using n! where n represents the number of objects to arrange or the order of events.

Example:

Toby, Jill, Ashley, and Mark are racing their bicycles. How many different ways can they finish the race?

Because four people are in the race, the different orders in which the four may finish is 4! = 4 x 3 x 2 x 1 = 24

Question 3

Permutation Problems With a Subset of the Entire Number of Objects or Events

Answer 3

Permutation problems become more involved when ou’re working with a subset of the entire number of objects or events, such as determining the possible number of ways three people out of 20 can finish a race first, second, and third or the total number of 4-digit sequences on a key pad that has 8 digits. To solve this type of permutation problem use the following formula:

P_rⁿ = n!

(n-r)!

Where,

P is the number of permuations you’re trying to determine

n is the total number of objects or events, and

r is the subset of objects or events you’re working with at one time

Example:

Ten high school soccer teams are competing for a berth in the upcoming tournament. How many possible ways can the top three teams be ranked from first to third?

Because order matters, this is a permutation problem. In this example, you have ten total teams, but you’re working with a subset of only three of those teams, so

n = 10 and

r = 3

P₃¹⁰ = 10!

(10 - 3)!

P₃¹⁰ = 10!

7!

P₃¹⁰ = 10 x 9 x 8 x 7!

7!

Reduce the fraction to its simplest terms first and then multiplying the remaining factors in the factorial, so cancel both 7! in the numerator and the denominator to get:

P₅²⁰ = 10 x 9 x 8 = 720

Question 4

When Order Doesn’t Matter: Combinations

Answer 4

A combination is a subset of objects or events in which order doesn’t matter; for example, choosing four pieces of fruit from a selection of 20 different types. If a question asks about choosing a number of items and the order in which items are arranged or events occur doesn’t matter, you’re looking at a combination problem.

To solve a combination problem use the following formula:

C^r_n = n!

r! (n-r)!

Where,

C is the number of combinations you’re trying to determine,

n is the number of objects or events, and

r is the number of objects or events you’re choosing.

Example:

From a group of ten colleagues, Sally must choose three to serve on a committee. How many possible combinations does she have to choose from?

Because the order doesn’t matter, this is a combination problem, so proceed as follows:

C³₁₀= 10!

3! (10-3)!

= 10 x 9 x 8 x 7!

3 x 2 x 1 x 7!

= 120

Question 5

Using Counting Methods to Solve Probability Problems