Which of the following show an element of the sample space for first rolling a die and then tossing a coin? Check all that apply. A.1T B.HH C.5H D.T8 E.6T F.34

Answer: When considering the sample space for first rolling a die and then tossing a coin, we need to look at the possible outcomes for each event and then combine them. There are six possible outcomes when rolling a die, which are the numbers 1 through 6. There are two possible outcomes when tossing a coin, which are heads (H) and tails (T). An element of the sample space would consist of a pair with the first component being the result of the die roll (a number from 1 to 6) and the second component being the result of the coin toss (either H or T).

Looking at the options given: A. 1T - This could represent rolling a 1 on the die and then getting tails on the coin toss, so it is a valid element of the sample space. B. HH - This does not represent a valid outcome since we are looking for an outcome from a die (a number) followed by an outcome from a coin (H or T). Both components here are coin outcomes. C. 5H - This represents rolling a 5 on the die and then getting heads on the coin toss, so it is a valid element of the sample space. D. T8 - The die only has numbers 1 through 6, so 8 cannot be the result of a die roll. Therefore, this is not a valid element of the sample space. E. 6T - This represents rolling a 6 on the die and then getting tails on the coin toss, so it is a valid element of the sample space. F. 34 - Since the die only has numbers 1 through 6, 34 cannot be the result of a die roll. Therefore, this is not a valid element of the sample space.