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Fundamental Counting Principle Overview The Fundamental Counting Principle determines the number of combinations one can make with data pulled from independent sets. For example, an ice‐cream store has many flavors of ice cream and many types of toppings. The flavors are elements in one set while the toppings are elements in another set. If you are allowed to choose one element from each set, you can easily determine how many combinations you can make by simply multiplying the number of elements in each set together! Thus, according to the fundamental counting principle, the number of possible combinations of data from independent sets can be determined by finding the product of the number of elements from each set. M x N = C M is the number of elements from one independent set N is the number of elements from another independent set C is the number of possible combinations from independent sets M and N This formula allows for more than two independent sets as well. For example, it is also true that, if M, N, and O are three independent sets then C = M × N × O. Take a look a the following example: An ice cream store offers 35 flavors and 15 toppings. If you must choose one flavor and one topping, how many combinations can you make? The ice cream flavors are a set, and the toppings are a set; therefore, you just need to find the product of 35 flavors and 15 toppings. 35 x 15 = 525 combinations, which is the correct answer. Fundamental Counting Principle with Independent Sets The Fundamental Counting Principle also works when you are making more than one choice from independent sets, but you must pay attention to how you are making those choices. If you can repeat choices made, then you count each set the number of times you are making choices. If you cannot repeat choices made, then you count each set the number of times you are making choices and deduct an element from the set each time. It’s like probability where you pull a marble out of a bag and put it back versus you pull a marble out of a bag and do not return it. Here is another example, with and without repeating choices: Take a look at the following example: An ice cream store has 20 flavors and 5 toppings. If you must choose two flavors and two toppings, how many combinations can you make? If you can repeat choices, then 20 flavors x 20 flavors x 5 toppings x 5 toppings = 10,000 combinations If you cannot repeat choices, then 20 flavors x 19 flavors x 5 toppings x 4 toppings = 7,600 combinations Keep this principle in mind when you are asked to find the combinations of sets. The Fundamental Counting Principle underlies other aspects of probability, including combinations and permutations. Practice Problems If an ice-cream store has 15 flavors, 10 toppings, and 2 containers, how many combinations of these items can be made if you must choose 1 of each? A hotdog stand has 4 types of wieners and 10 types of toppings. If you must choose 1 wiener and 3 different toppings (none can be repeated), how many combinations are possible? Your closet contains 10 pairs of jeans, 7 shirts, 8 pairs of different colored socks, and 12 pairs of shoes. If you select 1 article of clothing from each set, how many combinations are possible? In a restaurant you have 5 appetizers, 12 entrees, and 4 side dishes to choose from. How many possible combinations are possible if you choose 1 item from each set? Sally decides to repaint 2 walls in her home and has 12 different colors and 3 different painting methods to choose from. How many combinations can be made if Sally does not want the walls to be redone in the same way? a) 36 b) 72 c) 792 d) 1,296 At a recreation center, there are 5 basketball games, 7 coin games, and 9 shooting games. If you only have time for 3 games, how many combinations of games are possible if you play 1 of each type of game? a) 305 b) 315 c) 335 d) 345 A shaved ice place has 4 different flavors of ice and 10 different toppings. If you can only choose 1 flavor and 2 toppings, how many combinations can be made if you can have the same toppings for both topping choices? a) 40 b) 360 c) 400 d) 1,600 When building a car, you can choose from 3 colors, 3 interiors, 3 sound systems, 3 kinds of tires, and 3 seat coverings. How many combinations are possible if you are making 2 distinct cars? a) 7,776 b) 9,582 c) 11,079 d) 59,049 Answers to Practice Questions 300 2,880 6,720 240 C D B C
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