Suppose s is a set of 10 integers (any integers at all). prove that there are a pair of numbers in s whose difference is a multiple of 9; i. e. prove that there exists a, b ∈ s and an integer k ∈ z such that a − b = 9k. (hint: apply the pigeonhole principle to the potential remainders of elements of s with respect to division by 9).

The ratio of blue cards to green cards is 2 to 5. there are 8 blue cards. complete the diagram and explain how you can find the number of green cards.

Identify this conic section. x2 - y2 = 16 o line circle ellipse parabola hyperbola

Match the vocabulary to the proper definition. in probability, two events in which 1. conditional probability the outcome of one event is independent of the outcome of a second event. 2. dependent events in probability, two events in which the outcome of one event is dependent on the outcome of a second event. the probability of an event in which the outcome of the event is conditional on the outcome of one or more different events. 3. independent events