Combinatorics
How many ways can the letters in the word MATH be rearranged?
What is 4! or 24?
If A = {1, 2, 3} and B = {3, 4}, what is A ∪ B?
What is {1, 2, 3, 4}?
Define the domain of f(x) = 1/x.
What is all real numbers except 0?
You roll a fair 6-sided die. What is the probability of rolling a 4?
What is 1/6?
Max flips a fair coin and says, “If it’s heads, I’ll text Allura; if it’s tails, I'll text Will.” What’s the probability Abby ends up texting Will?
What is 1/2?
A pizza place has 4 toppings (pepperoni, mushrooms, onions, peppers). How many different 2-topping pizzas can be made if order doesn’t matter and no repeats are allowed?
What is C(4,2) = 6?
True or False: The set {∅} is the same as the empty set ∅.
What is False?
Is f: Z → Z defined by f(n) = n² one-to-one?
What is no?
A bag has 3 red and 5 blue marbles. You pick one without replacement. What is P(blue)?
What is 5/8?
Griffin loves Clash Royale and has 5 favorite cards: Giant, Knight, Wizard, Archer, Hog Rider. How many 2-card decks can he choose if order doesn’t matter?
What is C(5,2) = 10?
You flip a fair coin 5 times. How many possible sequences contain exactly 2 heads?
What is C(5,2) = 10?
Let U = {1,2,3,4,5,6,7}, A = {2,4,6}, and B = {1,2,3,4}. What is A′ ∩ B (where A′ is the complement of A in U)?
What is {1,3}?
Let f(n) = 2n and g(n) = n + 1. What is (f ∘ g)(3)?
What is 8?
(f(g(3)) = f(4) = 8)
A fair coin is flipped 3 times. What is the probability of getting at least one head?
What is 7/8?
Max creates a set M = {6, 7, Allura}. What is the power set P(M)?
What is P(M)= {}, {6}, {7}, {6,7}, {Allura}, {6, 7, Allura}, {6, Allura}, {7, Allura}?
How many different 5-digit “numbers” can you form using the digits 0–9 if the number must start with a nonzero digit and digits cannot repeat?
What is 9 × P(9,4) = 27,216?
How many subsets of {a, b, c, d, e, f} contain the element a?
What is 2^(6–1) = 32?
Suppose f: {1,2,3,4} → {a,b,c} is onto. How many onto functions exist?
What is 36?
Two cards are drawn without replacement from a standard 52-card deck. What is the probability that both are hearts?
What is 1/17?
((13/52)(12/51) = 1/17)
Will designs a function f: {King, Matt, Abby} → {6,7} where f(King)=6, f(Matt)=7, and f(Abby)=7. Is f onto? Is it one-to-one?
What is Onto but NOT one-to-one?
(Onto = Yes (both 6 and 7 are hit). One-to-one = No (Matt and Abby both map to 7).)
How many 4-letter “words” (not necessarily real words) can be made from the 26-letter alphabet if letters may repeat, but the word must contain at least one vowel (A, E, I, O, U)?
What is 262,495?
(Total = 26⁴ = 456,976. Words with no vowels = 21⁴ = 194,481. Subtract → 262,495)
Let X = {1,2,3,4,5}. How many ways can X be partitioned into exactly 2 nonempty subsets?
What is S(5,2) = 15?
Let f: R → R be defined by f(x) = x³ – 2x. Show whether f is one-to-one and/or onto.
What is one-to-one and onto?
(f is one-to-one (strictly increasing cubic, passes horizontal line test) and onto (as x→±∞, f(x)→±∞, covering all reals).)
You roll two fair dice. What is the probability that the sum is divisible by 3?
What is 1/3?
(There are 36 outcomes. Sums divisible by 3: {3,6,9,12}. Count = 12 outcomes. Probability = 12/36 = 1/3)
In Clash Royale, suppose King wants to choose a 3-card hand from a pool of 8 cards, but one of them must always be “7” (his favorite card number). How many different hands can he build?
What is C(7,2) = 21?
(Fix card “7” in every hand. Then choose 2 from the remaining 7 cards. = C(7,2) = 21)