What is the formula for the number of permutations of n objects taken r at a time?

P(n, r) = n! / (n - r)!

True or False: The number of circular permutations of n different objects is (n-1)!.

True.

Riddle: I can arrange n objects in a straight line, but when arranged in a circle, my count decreases. What am I?

Circular permutations.

The number of permutations of n different objects taken r at a time when p particular objects do not occur is represented as ______.

N - nPr

Which of the following represents combinations of n objects taken r at a time? A) P(n, r) B) C(n, r) C) n! D) nPr

B) C(n, r)

True or False: The pigeonhole principle states that if n + 1 objects are put into n boxes, at least one box must contain exactly one object.

False. It states that at least one box must contain at least two objects.

What is the generating function for the constant sequence where every term is equal to 1?

G(x) = 1 / (1 - x)

Fill in the blank: A sequence defined by αn = n² for all n is known as ______.

Square number sequence

For a linear recurrence relation of the form Fn = AFn−1 + BFn−2, what is the characteristic equation?

X² - Ax - B = 0

Riddle: I'm a method used to solve recurrence relations and can represent sequences as coefficients of powers. What am I?

Generating function.

True or False: The number of ways to fill r places with n elements when repetition is allowed is n^r.

True.

What is the formula for the number of permutations of n objects taken r at a time?

P(n, r) = n! / (n - r)!