Fundamental Counting Principle — Tree Diagrams, Flipping a Coin, Criminology, License Plates, Menus, Passwords

When there are m ways to do one thing, and n ways to do another thing, the there are m x n ways of doing both. This is called the fundamental counting principle and requires multiplication or repeated addition.

With grades 1-2, I taught the children to create tree diagrams which is an easy pictorial representation of the number of ways or combinations of certain outcomes. First, we flipped a coin (heads or tails gives you two ways), then a second coin (four ways — HH, HT, TH, TT), then a third coin (2 x 2 x 2 = 8 ways), and so on. The attached pdf allows them to explore doubling for multiple flips of a coin. 

Clothes Combinations:

We also looked at the number of combinations of choosing clothes to wear. With all of the classes, I shared with them a very simplistic lifestyle where a person might have 5 pairs of pants, 4 shirts, and 3 pairs of shoes, giving them 5 x 4 x 3 = 60 ways of choosing one pair of pants, one shirt and one pair of shoes. The children shared with me how many pants, shirts and shoes they have and some of the combination numbers were staggering. For example, one girl had 50 pair of pants, 40 shirts and 30 pair of shoes, conveniently, we took the non-zero numbers of 5 x 4 x 3 = 60 and tacked on the three zeros to get 60,000 combinations. Of course, we are assuming that every pant, shirt, or shoes match; realistically, they would not, but for simplicity, this is an extraordinary number of choices for anyone to make. It would be a fun exercise for you and your child to go through their wardrobe and count the number of pants, shirts and shoes to go through a fundamental counting principle calculation. If the numbers are in the double digits and not multiples of ten, they should round up or down to the nearest ten (44 rounds down to 40 and 45 rounds up to 50).

Meal Combinations:

I also seek restaurants with few choices on the menu because I know that the more combinations a restaurant has to anticipate, the less fresh is the food. Case in point, the Cheesecake Factory has 41 Drink items, 49 Appetizers, 155 Entrees, and 44 Desserts. If you order one drink, one appetizer, one entree, and one dessert, the number of combinations total 13,701,380. A quick calculation of the fundamental counting principle would be to estimate 40 x 50 x 160 x 40. First start with multiplying 4 x 5 x 16 x 4 to get 1280 and then tack on the four zeros to get 12,800,000 combinations; pretty close to the actual number. 

Criminology: 

Witness identification kits have 195 hair/hairlines, 99 eyes/eyebrows, 89 noses, 105 mouths, and 74 chins/cheeks. These kits can identify close to double the number of people on the planet — 13,349,986,650. Using the power of estimating, if you multiply 200 x 100 x 90 x 100 x 70, and first multiply the non-zero numbers of 2 x 1 x 9 x 1 x 7 = 126, then tack on eight more zeros you will get 12,600,000,000, very close to the actual number of combinations. If witnesses can narrow down a few of these features such as hair color and eyes, it makes the job easier for the police because the fundamental counting principle yields far fewer combinations.

License Plates:

I gave the children examples of each state license plate. In New York, they require three digits followed by three letter, and since you can only use capital letters and may repeat digits and letters, the calculation is 10 x 10 x 10 x 26 x 26 x 26 = 17,576,000 and if no repeat is allowed, then 10 x 9 x 8 x 26 x 25 x 24 = 11,577,600. We had a discussion of the fact that New York state has over 15 million people many of whom have more than one car. The children took a while to realize that not everyone in the state owns a car, especially children from 0-16 years old.

Passwords:

The richest discussion was regarding passwords as our QWERTY keyboard has 94 characters available for passwords. To my knowledge, emojis are not yet used in passwords. There are 10 digits, 26 upper case and 26 lower case letters and 32 additional characters such as ~!@#$%^&*()_+`-={}|:”<>?[]\;’,./. So a 6 character password can be up to 94^6 different combinations or 689,869,781,056. Using estimating by multiplying 100 x 90 x 100 x 90 x 100 x 90 = 729 and tack on 6 zeros for 729,000,000,000, again pretty close to the actual number.

We also discussed fun ways to create memorable passwords by chosing a word such as MATH and holding the shift key while pressing on the numbers above (slightly to the right) the digits MATH which would be the numbers 9, 2, 6, and 7 which would give you a password of (@^&. This would be difficult to hack.

Have fun looking at the world through fundamental counting principle glasses. And try to have a simpler new year.

The pdfs with solutions are attached below.