Creepy Combinatorics
How many ways are there to place n rooks on an n-by-n chessboard so that no two rooks could attack one another? (I.e., no pair of rooks share a row or column of the board.)
n!
Determine the value the following series converges to:
\sum_{n=0}^\infty \frac{2 * 5^n + 4 * 3^n}{15^n}
8
What is the prime factorization of 2240?
2^6 * 5 * 7
Given a rectangle with base of 6 and height of 8, if we take the midpoints of each side and make a new figure from those points, what is the area of the new figure?
8
A random card is drawn from a shuffled deck of 52 playing cards. What is the probability the card is a king or a heart?
A standard deck of playing cards has 4 suits (diamonds, clubs, hearts, spades) and each suit has one card of each of 13 ranks (2, 3, …, 10, Jack, Queen, King, Ace).
Pr(K∪H) = Pr(K) + Pr(H) - Pr(K∩H) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13, or about 0.3077.
How many distinct 11-character strings can be constructed by permuting the letters MATHEMATICS?
\frac{11!}{2! * 2! * 2!} = \frac{11!}{8} = 4989600
Determine the interval of convergence for the power series
\sum_{n=0}^\infty \frac{x^n}{n!}
(-\infty, \infty)
What is the value of
cos(tan(sin(cos(pi/2))))
1
Given a right triangle with sides of lengths 5, 12, and 13, what is the radius of the circumcircle of the triangle?
13/2
Sore throat is present in 70% of people with the flu, and present in 15% of people without the flu. If 10% of people in a given population have the flu, what is the probability a randomly selected person has a sore throat?
Let E be the event someone has sore throat, and F is the event someone has has the flue. The law of total probability states
Pr(E) = Pr(E|F)Pr(F) + Pr(E|F^c)Pr(F^c)
The probability a randomly selected person has a sore throat is then
Pr(E) = 0.7 * 0.1 + 0.15 * 0.9 = 0.205
A certain Congressional committee consists of 10 members, of whom 6 are Democrats and 4 are Republicans. How many ways can a 6-person subcommittee consisting of exactly 3 Democrats and 3 Republicans be chosen?
((6),(3)) * ((4),(3)) = 80
What is the value of the following series?
\sum_{n=0}^\infty 3^{2+n} 2^{1-3n}
\sum_{n=0}^\infty 3^{2+n} 2^{1-3n}=\sum_{n=0}^\infty 18 (3/8)^n = \frac{18}{1 - 3/8} = 144/5
What are all solutions to the equation
x + y = z
modulo 2?
1+0=1
0+1=1
0+0=0
1+1=0
Given a cube with a volume of 216, what is the volume of the largest possible sphere contained inside of the cube?
36\pi
Suppose the probability that a certain basketball player scores on a free throw is 0.7. If the player shoots ten free throws, what is the probability the score on exactly eight of those free throws?
((10),(8)) * (0.7)^8 * (0.3)^2 \approx 45 * 0.0576 * 0.09 \approx 0.2333
Suppose P1, P2, …, P8 are eight distinct points in the plane, and exactly three of these points are collinear. How many (non-degenerate) triangles can be formed using three of these points as vertices?
((8),(3)) - ((3),(3)) = 56 - 1 = 55
What is the radius of convergence of the following power series?
\sum_{n=1}^\infty \frac{(3x + 9)^n}{n}
3
What are all of the solutions to
x^3 - 8 = 0
2
-1-i\sqrt{3}
-1+i\sqrt{3}
Suppose ABCD is parallelogram of area 34 with bases AB and CD having a length of 4. If AB lines on a line L1 and CD lies on a line L2, and if we shift CD five units along L2, what is the area of the resulting parallelogram?
34
Suppose the number of typos on a single page of a certain book is, on average, one typo per page. Modelling the number of typos by a Poisson random variable, what is the probability a given page has at least one typo?
The number of typos is modeled by a random variable X which is Poisson with parameter … and so the probability of at least one typo is
Pr(X \geq 1) = 1 - Pr(X = 0) = 1 - 1/e ~~ 0.6321
Suppose that license plates in a certain state must follow be given by three letters, followed by a dash, followed by four digits. How many possible license plates are there?
26^3 \cdot 10^4 = 175760000
What power series centered at 0 equals the following function where the series converges?
\frac{5}{1 - 4x^2}
5 \sum_{n=0}^\infty 4^n x^{2n}
Given a function
f(x) = 2x^5 + 4x^3 + 2x - 100
Find all real values in the preimage
f^{-1}({0})
f^{-1}(\{0\}) \cap \mathbb{R} = \{2\}
Suppose four points lie on a collection of lines subject to the following conditions: any two distinct points lie on exactly one line, and every line contains exactly two distinct points. How many lines are there?
6
Taking turns, two people play game where they flip a fair, two-sided coin. If the coin lands on heads, whoever flipped it wins the game. If the coin lands on tails, the other person gets to flip the coin.
What is the probability the first player will win the game?
The probability of first getting heads on the n-th flip is 1/2^n. The first player wins if the first heads occurs on an odd-numbered flip, and so the probability the first player wins is
\sum_{k=1}^\infty (1/2)^(2k-1) = 2/3