Basic High-School Math Flashcards
Counting principle, Permutations and Combinations, Palindrome, Subsets of a set, Basic probability, Sum of first N natural numbers, Slope of a line and derivative, Distance Formula, Prime numbers, Pigeonhole principle, LCM and GCD, Inclusion Exclusion principle, Decimal and Binary, Hexadecimal, Solving a quadratic equation, Mean, Median, Mode (23 cards)
The Counting Principle
When there are m ways to do one thing, and n ways to do another, there are m * n ways to do both.
https://www.youtube.com/watch?v=s_LfN4ItCs4
Permutations with reputition
n^r
n is the number of things to choose from
r is how many times you choose
e.g. For a lock with numbers 1 through 10, n would be 10
If the lock had 5 slots, r would be 5.
10^5 = 100,000 possible codes
https://www.youtube.com/watch?v=s_LfN4ItCs4
Permutations without reputition
n!/(n-r)!
“n factorial over n minus r factorial”
Shorthand: n P r
n is the number of things to choose from
r is how many times we can choose
e. g. If you need to choose four unique letters in the alphabet, you start out with 26 letters to choose for the first slot. But then only 25 for the second, 24 for the third, and 23 for the fourth. So you’d have 26 factorial divided by 26-4 (22) factorial choices which calculates out to 358,800 permutations without reputition.
https: //www.youtube.com/watch?v=s_LfN4ItCs4
How are combinations different than permutations?
Permutations are different with each order of choices, combinations are the same no matter the order.
e. g. 1, 2, 3 is a different permutations of 3, 2, 1, but still the same combination
https: //www.youtube.com/watch?v=s_LfN4ItCs4
Combinations without reputition
n! / r! (n-r)!
“n factorial over r factorial times n minus r factorial”
There will always be r factorial more permutations than combinations, so you just take the permutations without reputition and multiple the part you are dividing by, by r factorial.
e.g. If there are 3 slots with numbers from 1 - 3 then there are 3! aka 6 more permutations than combinations…
Possible permutations without reputition: 6 1, 2, 3 2, 3, 1 3, 2, 1 2, 1, 3 1, 3, 2 3, 1, 2
n! / (n-r)!
3! / (3-3)!
6 / 1
6 permutations without reputition
Combinations without reputition: 1
1, 2, 3
n! / r! (n-r)!
3! / 3! (n-r)!
6 / 6 * 1
1
https://www.youtube.com/watch?v=s_LfN4ItCs4
Factorial
n!
Afactorialis a function that multiplies a number by every number below it. For example 5!= 54321=120. The function is used, among other things, to find the number of ways “n” objects can be arranged.
What is a Palindrome? Examples?
A word (or sentences, or groups of numbers, or equations) spelled the same backward and forward.
mom, dad, poop, ewe, noon, peep, racecar, redder, madam, kayak
Madam, I'm Adam. I did, I did? He did, eh? Never odd or even. Oh no, Don Ho! Step on no pets. Too hot to hoot. Yo! Banana Boy!
2002, 1991, 1771, 100,001, 111,111,111 x 111,111,111, 12,345,678,987,654,321
https://www.youtube.com/watch?v=FIvONwfz9Rw
What is a proper subset? What is the mathematical symbol for a proper subset?
A set with less elements than its superset.
Symbol is a sideways U… pointing to the right.
https://www.youtube.com/watch?v=s8FGAclojcs
When is a subset not proper? What is the mathematical symbol for it?
When the set includes exactly the same elements as its superset.
The symbol is a sideways U with a line under it.
How do you write “the probability of A or B”?
How do you write “the probability of A and B”?
P(AuB)
P(AnB)
https://www.youtube.com/watch?v=Y8EOkJ75elM
What is conditional probability?
How do you write it with event A and B?
The probability of an event occuring on the condition that another event has already occured.
P(A|B)
How do you calculate the probability with disjoint/mutually exclusive events?
Addition rule… add the two probabilities together.
Disjoint/mutally exclusive means event A and event B cannot occur together… so you can add their probabilities together.
If the probability of A is 20% and the probability of Be is 30% then the probability of A or B occuring is 20+30=50%
How do you calculate the probability of two events that can occur together?
P(A) + P(B) - P(AnB)
P(A): 20%
P(B): 30%
P(AnB): 10%
20+30-10= 40%
How do you calculate the probability of two events both occuring if the events are independent?
If events are independent (Event A does not Affect B) use Multiplication rule: P(A|B) = P(A) * P(B)
How do you calculate the probability of two events both occuring if the events are NOT independent?
If the events affect each other use The Multiplication rule: P(A) * P(B|A)
“Probability of A times probability B on the condition that A already occured”
How do you calculate conditional probability?
P(A|B) = P(AnB)/P(B)
“Probability of A occuring on the condition of B already occuring is equal to the probability of A and B divided by the probability of B.”
P(A): 20%
P(B): 45%
P(AnB): 15%
P(A|B): 15/45 = 33%
What does the Pigeon Hole Principle tell us?
That if you have n number of “pigeons”, but only n-1 “holes” then at least two “pigeons” must share a “hole”.
Shorthand: PHP
Uses:
Given a set of numbers {3,4,5,6,7,8,9,10,11,12}
Pick 6 numbers. At least two of those numbers will add up to 15. This is because each number in the set pairs with another number to add up to 15. This means there are 5 “holes” (10 numbers paired up is 5 holes). If you pick more than 5, like in this case where we are picking 6, then we can trust PHP to know that at least two numbers will sum to 15.
https://www.youtube.com/watch?v=ROnetLvbl6M
What is the Inclusion/Exclusion principle used for?
It is used to avoid over counting.
Imagine a ven diagram. This principle would help you avoid counting anything in the intersection of the two categories more than once.
A + B - AnB
“A plus B minus A intersection B”
https://www.youtube.com/watch?v=B8mOws755Bs
What is the pattern for finding the formula for Inclusion/Exclusion principle?
If you have n categories then you sum n categories then you subtract the intersection of each pair of categories then you add the intersection of each triplet of categories and continue this cycle (adding then substracting) for a total of n cycles.
For example, imagine a ven digram with 5 categories (A,B,C,D). You’d count them with this forumla:
(A+B+C+D) - (AnB - AnC - AnD - BnC - BnD - CnD) + (AnBnC + AnBnD + AnCnD + BnCnD) - AnBnCnD
Note AnB is pronounced “A intersection B”
Should actually be written like (4 1) - (4 2) + (4 3) - (4 4)
Except with the second number in each parenthesis actually below the first.
Pronounced: “4 choose 1 minus 4 choose 2 plus 4 choose 3 minus 4 choose 4”
https://www.youtube.com/watch?v=B8mOws755Bs
How do you convert base 10 to Binary?
1024 | 512 | 256 | 128 | 64 | 32 | 16 | 8 | 4 | 2 | unit
| | | | | | | | | |
8421
1: 0001
2: 0010
3: 0011
4: 0100
5: 0101
6: 0110
7: 0111
8: 1000
9: 1001
10: 1010
11: 1011
12: 1100
13: 1101
14: 1110
15: 1111
https://www.youtube.com/watch?v=CiJkiZJOJCk
How do you convert Base 10 to Hexadecimal?
4096|256|16| Unit
0: 0 0 0 0
1: 0 0 0 1
2: 0 0 0 2
3: 0 0 0 3
…
10: 0 0 0 A
11: 0 0 0 B
12: 0 0 0 C
13: 0 0 0 D
14: 0 0 0 E
15: 0 0 0 F
16: 0 0 1 0 161
17: 0 0 1 1 161+1
18: 0 0 1 2 161+2
19: 0 0 1 3 161+3
20: 0 0 1 4 161+4
…
25: 0 0 1 9 161+9
26: 0 0 1 A 161+10
27: 0 0 1 B 161+11
28: 0 0 1 C 161+12
29: 0 0 1 D 161+13
30: 0 0 1 E 161+14
31: 0 0 1 F 161+15
32: 0 0 2 0 162+0
33: 0 0 2 1 162+1
34: 0 0 2 2 162+2
35: 0 0 2 3 162+3
…
41: 0 0 2 9 162+9
42: 0 0 2 A 162+10
43: 0 0 2 B 162+11
44: 0 0 2 C 162+12
45: 0 0 2 D 162+13
46: 0 0 2 E 162+14
47: 0 0 2 F 162+15
48: 0 0 3 0 163
…
144: 0 0 9 0 169
145: 0 0 9 1 169+1
146: 0 0 9 2 169+2
147: 0 0 9 3 169+3
…
153: 0 0 9 9 169+9
154: 0 0 9 A 169+10
155: 0 0 9 B 169+11
156: 0 0 9 C 169+12
157: 0 0 9 D 169+13
158: 0 0 9 E 169+14
159: 0 0 9 F 169+15
160: 0 0 A 0 1610
161: 0 0 A 1 1610+1
162: 0 0 A 2 1610+2
…
169: 0 0 A 9 1610+9
170: 0 0 A A 1610+10
171: 0 0 A B 1610+11
172: 0 0 A C 1610+12
173: 0 0 A D 1610+13
174: 0 0 A E 1610+14
175: 0 0 A F 1610+15
176: 0 0 B 0 1611
177: 0 0 B 1 1611+1
…
185: 0 0 B 9 1611+9
186: 0 0 B A 1611+10
…
191: 0 0 B F 1611+15
192: 0 0 C 0 1612
…
240: 0 0 F 0 1615
…
249: 0 0 F 9 1615+9
250: 0 0 F A 1615+10
255: 0 0 F F 1615+15
256: 0 1 0 0 2561
257: 0 1 0 1 2561+1
…
271: 0 1 0 F 2561+15
272: 0 1 1 0 (2561)+(161)
273: 0 1 1 1 (2561)+(161)+1
274: 0 1 1 2 (2561)+(161)+2
…
287: 0 1 1 F (2561)+(161)+15
288: 0 1 2 0 (2561)+(162)
289: 0 1 2 1 (2561)+(162)+1
…
496: 0 1 F 0 (2561)+(1615)
511: 0 1 F F (2561)+(1615)+15
512: 0 2 0 0 (2562)
…
3840: 0 F 0 0 (25615)
…
4095: 0 F F F (25615)+(1615)+15
4096: 1 0 0 0 (40961)
…
61440: F 0 0 0 (409615)
…
65535: F F F F (409615)+(25615)+(16*15)+15
A=10 B=11 C=12 D=13 E=14 F=15
0
https://www.youtube.com/watch?v=4EJay-6Bioo
What is the Quadratic Equation and how do you use it?
x= (-b +- √b^2 - 4ac) / 2a
“x equals negative b plus or minus the square root of b squared minus four a c over two a”
Use it to solve for x for standard equations in this form: ax^2 + bx + c = 0
Example:
2x^ + 2x - 5 = 0
a=2 b=2 c=-5
x = (-2 +- √2^2 - 4*2*-5) / 2*2 x =( -2 +- √4 - 4*-10) / 4 x = (-2 +- √4 + 40) / 4 x = (-2 +- √4*√11) / 4 x = (-2 +- 2*√11) / 4 x = (-1 +- √11) / 2
https://www.youtube.com/watch?v=3ayhvAI3IeY
What’s the difference between mean, median, and mode?
Mean is the average of a set.
Median is the middle number in a set.
Mode is the number that shows up the most in a set.
{23, 29, 20, 32, 23, 21, 33, 25}
Mean: (23 + 29 + 20 + 32 + 23 + 21 + 33 + 35 + 25)/8 = 25.75
Median: 20, 21, 23, 23 | 25, 29, 32, 33 = (23+25)/2 = 24
Note: Ordered then split in half. If even then take to middle numbers and divide their sum.
Mode: 23
