Problem 1
In how many ways can \(n\) people be seated around a round table? (Remember that two seating arrangements around a round table are equivalent if everyone is in the same position relative to everyone else in both arrangements.)
Problem 1
All problems with blue boxes have an answer or hint available at the end of
the book.
The following segment of code is part of a program that uses insertion sort to
sort a list \(A\).
for \(i=2\) to \(n\)
\(j=i\)
while \((j \geq 2)\) and \((A[j]
Problem 1
Find \(\left(\begin{array}{c}12 \\ 3\end{array}\right)\) and \(\left(\begin{array}{c}12 \\ 9\end{array}\right)\). What can you say in general about \(\left(\begin{array}{l}n \\ k\end{array}\right)\) and \(\left(\begin{array}{c}n \\ n-k\end{array}\right) ?\)
Problem 2
Find the row of the Pascal triangle that corresponds to \(n=8\).
Problem 2
List all the functions from the three-element set \(\\{1,2,3\\}\) to the set \(\\{a, b\\}\). Which functions, if any, are one-to-one? Which functions, if any, are onto?
Problem 2
Five schools are going to send their baseball teams to a tournament in which each team must play each other team exactly once. How many games are required?
Problem 2
Determine whether the following relations are equivalence relations. a. "Is a brother of or is" on the set of people b. "Is a sibling of or is" on the set of people c. " \(x\) is related to \(y\) if \(|x-y| \leq 2\) " on the set of integers
Problem 3
Use binomial coefficients to determine the number of ways in which you can line up three identical red apples and two identical golden apples. Use equivalence class counting (in particular, the quotient principle) to determine the same number.
Problem 3
In how many ways can you draw a first card and then a second card from a deck of 52 cards?
Problem 3
Find the following a. \((x+1)^{5}\) b. \((x+y)^{5}\) c. \((x+2)^{5}\) d. \((x-1)^{5}\)
