To determine the sample space for the experiment of selecting a coin from a bag containing a nickel (N), a quarter (Q), and a dime (D) three times (with replacement), we need to consider all the possible combinations of selections.
Since there are 3 coins and each selection is independent (due to the replacement), we can represent the sample space using ordered triples (where each position in the triple corresponds to one selection). Each position can be filled with either N, Q, or D.
The total number of possible outcomes is given by \(3^3\) (since there are 3 choices for the first selection, 3 for the second selection, and 3 for the third selection), resulting in \(27\) outcomes.
The sample space consists of all combinations:
- For the first position, you can select either N, Q, or D.
- The same options apply for the second and third positions.
Thus, the sample space includes the following combinations:
- NNN
- NNQ
- NND
- NQN
- NQD
- NQQ
- NDN
- NDQ
- NDD
- QNN
- QNQ
- QND
- QQN
- QQD
- QDQ
- QDN
- QQQ
- QDQ
- QDD
- DNN
- DNQ
- DND
- DDQ
- DQD
- DQN
- DQQ
- DDD
Regarding the options provided:
- Option 1 and Option 2 both include incorrect strings and invalid combinations.
- Option 3 has repeated elements and also invalid combinations.
- Option 4 lists all the correct combinations without any repetitions or missing combinations.
Therefore, the correct answer is:
4. NNN, NNQ, NND, NQN, NQD, NQQ, NDN, NDQ, NDD, DDD, DDQ, DDN, DQD, DQN, DQQ, DND, DNQ, DNN, QQQ, QQD, QQN, QDQ, QDN, QDD.