\begin{tabular}{|l||}\multicolumn{1}{l|}{ Permutations and Probability } \ \hline \begin{tabular}{l} Determine the following probabilities. Enter your answers as percents rounded to four decimal \ places. \ \hline Dwayne is creating a 7 digit passcode using the digits 0 through 9 . \ If Dwayne chooses all of the characters at random, what is the probability that no digits are repeated? \ \hline Three cards are chosen without replacement from a deck of 52 cards and placed in order from first to \ third. \ What is the probability that all three cards are $$ 6 s $$ ? \ \hline\end{tabular} \ \hline\end{tabular}

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### **Problem 1: 7-Digit Passcode with No Repeated Digits**

**Total Possible Passcodes:**
Each of the 7 digits can be any of the 10 digits (0-9), so:
$$ 10^7 = 10,000,000 $$

**Passcodes with No Repeated Digits:**
For the first digit, there are 10 choices, for the second digit 9 choices (since it must be different), for the third digit 8 choices, and so on:
$$ 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 = 604,800 $$

**Probability:**
$$ \text{Probability} = \left( \frac{604,800}{10,000,000} \right) \times 100 = 6.0480\% $$

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### **Problem 2: Drawing Three 6s in Sequence from a Deck**

**Total Possible Sequences:**
There are 52 cards, so the number of ways to draw three cards in order is:
$$ 52 \times 51 \times 50 = 132,600 $$

**Favorable Sequences (All Three Cards Are 6s):**
There are four 6s in a deck. The number of ways to draw three distinct 6s in order is:
$$ 4 \times 3 \times 2 = 24 $$

**Probability:**
$$ \text{Probability} = \left( \frac{24}{132,600} \right) \times 100 \approx 0.0181\% $$

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**Final Answers:**

1. **Probability of no repeated digits in the passcode:**  
   **6.0480%**

2. **Probability that all three drawn cards are 6s:**  
   **0.0181%**
- **Probability of no repeated digits in the 7-digit passcode:** 6.0480% - **Probability that all three drawn cards are 6s:** 0.0181%

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