Final answer:
Using the combination formula, the number of outcomes for picking 3 numbers without considering order can be determined, but the method is not fair for picking a historian as it does not meet the criteria for binomial probability and does not ensure equal chances for each individual.
Step-by-step explanation:
To determine the number of outcomes for picking 3 numbers without considering order, you use the combination formula, which is employed for situations where order does not matter. This is known as a combination as opposed to a permutation, where order would be significant. The number of combinations of n objects taken r at a time is given by the formula:
C(n, r) = n! / [r!(n - r)!],
where 'n' is the total number of items, 'r' is the number of items to choose, and '!' denotes factorial. However, the original question asks about the fifth counting rule, which is often associated with the binomial theorem and refers to the probability of an event occurring in a sequence of independent trials. This is typically used when there is a fixed number of trials and only two possible outcomes (success or failure) in each trial.
In terms of picking a historian, if the statement is referencing selecting a historian at random using probability and counting rules, this would not be fair as it does not involve repeated trials or only two outcomes (the historian is either picked or not). The key aspects of fairness in selection would involve every historian having an equal chance of being picked, which is more aligned with random sampling.
Answer to the sub-question:
'b. Suppose you know that the picked cards are Q of spades, K of hearts, and Q of spades. Can you decide if the sampling was with or without replacement?'
The presence of the same card, the Queen of spades, appearing twice indicates that the sampling was with replacement because the same item was picked more than once. Without replacement, each card could only appear once.