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Fundamental Counting Principle
Permutation Applications
More Permutation Applications
Combinations
Basic Probability
100
Dave is picking classes to take next year in High School. There are 3 math classes and 6 science classes. How many different sets of classes can Dave choose?
18 different choices m1*m2 = 3*6 = 18
100
What does n represent?
n represents the total number of objects from the set that you are picking from
100
What does r represent?
r represents the number of objects you are picking from the set
100
What is the definition of a Combination?
A method for selecting subsets of a larger set, where the order is not important.
100
What is the definition of Sample Space?
The set of all possible outcomes of the of an experiment or trial.
200
Emma goes to a car part yard to get parts for the car she is building. She can choose 1 of 6 engines, 1 of 12 fuel filters, and 1 of 7 rear-view mirrors. How many car part sets could Emma have for her car?
504 sets 6*12*7 = 504
200
How many permutations are possible from the set of letters Q, R, S, T, U, V, W?
5040 7 letters n! = 7! = 7*6*5*4*3*2*1 = 5040
200
What is the definition of a Permutation?
A method for selecting and ordering subsets of a larger set, where the order is important
200
Solve 12C4 using the combination formula.
495 nCr = n!/(n-r)!r! 12C4 = 12!/8!4! = 12*11*10*9/4*3*2*1 = 495
200
If a gambler plays a few games of slots, and then he says, "Because I lost, in the next few rounds I'm more likely to win!" Is he correct? Why or why not?
He is not correct. In probability, each event is independent; unless the machine remembers his previous trials, the probability that he wins remains the same every time he plays.
300
If you flip a coin 12 times, how many sequences are possible? (if the coin either lands on heads or tails)
4096 possible sequences (Heads, Tails) = 2 outcomes of flipping the coin 2*2*2*2*2*2*2*2*2*2*2*2 = 4096
300
Evaluate 19P3 using the permutation formula
5814 nPr = n!/(n-r)! 19P3 = 19!/(19-3)! = 19!/16! = 19*18*17*16!/16! = 19*18*17 = 5814
300
How many distinguishable ways can the word LLAMA be arranged?
30 unique ways 5 = total letters Given the distinguishable permutation formula, n = n1+n2+n3+...+nk 5 = 2+2+1 5!/2!*2!*1! = 5*4*3/2*1*1 = 30
300
Solve 27C7 using the combination formula.
888,030 27C7 = 27!/20!7! = 27*26*25*24*23*22*21/7*6*5*4*3*2*1 = 888,030
300
I roll two 6-sided dice. What is the probability that I roll a 10 or higher?
Sample Space = 36 outcomes Successful outcomes = 6 (6+4, 4+6, 5+5, 5+6, 6+5, 6+6) 6/36 = 1/6 P = 1/6
400
If robbers are breaking in to a bank, and there are 2 locks with 6 digit number sequences (where numbers can repeat) on the safe, what is the total number of combinations the robbers need to try on the safe?
1,000,000 combinations 10*10*10*10*10*10 = 1,000,000
400
Evaluate 122P4 using the permutation formula
210,801,360 122P4 = 122!/(122-4)! = 122!/118! = 122*121*120*119*118!/118! = 122*121*120*119 = 210,801,360
400
How many distinguishable ways can the word PRODUCT be arranged if the letter D always comes before U?
2520 arrangements Since no letters repeat = 7! Since 2 letters are already predefined, you are taking away a factor of 2 7!/2! = 7*6*5*4*3 = 2520
400
There are 9C4 ways of choosing 4 books from a shelf and 11C6 ways of choosing books from the floor. How many combinations of books can you take?
58,212 Combine using Fundamental Counting Principle 9C4*11C6 = 9!/5!4!*11!/5!6! =126*462 = 58,212
400
In a park, there are 398 oak trees, 99 birch trees, 178 maple trees and 50 spruce trees. If a random tree is selected from the park, what is the probability it will be a maple tree? (Write all forms of answer)
P = 178/725, 0.2455, 24.55%.
500
Minneapolis has 3 area codes for phone numbers. If you're trying to find the rest of the phone number (the 7 digit number after the area code), how many possibilities could Minneapolis have for telephone numbers? (The first number of the 7 digit number cannot start with a 0 or a 1)
24,000,000 phone numbers 3 = area codes 8 = possibilities for first digit of 7 digit number 10^6 = rest of possibilities for the 7 digit number 3*8*(10^6) = 24,000,000
500
95 kids from the class of 2019 are all competing to place first in the Spelling Bee. Using the permutation formula, find all possible outcomes of placing in the first 5 places.
6,952,862,280 95P5 = 95!/(95-5)! = 95!/90! = 95*94*93*92*91*90!/90! = 95*94*93*92*91 = 6,952,862,280
500
I have a 2x2x2 Rubik's cube that has 8 pieces, and since they are all corner pieces they have 3 stickers on each piece. I accidentally dropped it on the floor, and the pieces scattered everywhere. How many ways can I reconstruct the cube?
88,179,840 Order of pieces*Orientation of stickers 8 cubes = 8! 8!*3^7 (3^7 because of reference point) =88,179,840
500
You need to make a box of 10 chocolates. 5 milk, 3 dark, and 2 white. You have 15 milk to choose from, 8 dark, and 6 white. How many different boxes of chocolates are possible?
2,522,520 15C5*8C3*6C2 = (15!/10!5!)*(8!/5!3!)*(6!/4!2!) = 3003*56*15 = 2,522,520
500
There are 400 people in an experiment. 37 people were given no treatment. 127 people were given Treatment A, and 240 people were given Treatment B. A few individuals were given both treatments. What is the probability that a given individual got a treatment?
P = (400 - 37) / 400 = 0.9075. There is a 90.75% chance an individual got a treatment.