Question

Prove or disprove each statement:

(a) If n is an even integer, then ________.
(b) If n is an odd integer, then ________.
1) The solution is even
2) The solution is odd
3) The solution is prime
4) The solution is composite

Answer 1

The question relates to the properties of odd and even numbers and calculating probabilities using two dice, as well as understanding the Binomial theorem for algebraic expansion.

Step-by-step explanation:

Probability Statements

The question seems to be a mix of probability and basic number properties. The initial part of the question (a) and (b) related to odd and even numbers seems to be incomplete, so let's address the remaining parts of the question.

Probability and Dice Rolls

When determining probabilities of events with dice, you need to consider the sample space and whether the events are independent or dependent. For example, Event A being that both dice show an even number has a probability determined by the count of even numbers available on the dice, and Event B that both show a number greater than eight involves the numbers 9, 10, 11, and 12. If two 12-sided dice are involved, you can calculate specific probabilities by counting the favorable outcomes and dividing by the total outcomes.

The Binomial theorem part of the question provides the expansion of the binomial expression (a + b) raised to the power of n, which is a fundamental concept in algebra.