The problem can be summarized as follows:

Question: From the digits $\{0, 1, 2, 3, 4, 5, 6\}$, three are drawn without replacement. Calculate the probability that the number formed by these digits is divisible by $5$.

Solution:

A number is divisible by $5$ if and only if its last digit is $0$ or $5$. We calculate the probability step by step:

Total number of possible outcomes

The total number of ways to select $3$ digits from the $7$ available is: $\binom{7}{3} = \frac{7 \cdot 6 \cdot 5}{3 \cdot 2 \cdot 1} = 35$ For each combination of $3$ digits, the digits can be arranged in $3! = 6$ different ways to form numbers. Thus, the total number of three-digit numbers is: $35 \cdot 6 = 210$

Favorable outcomes (numbers divisible by $5$)

1. Case 1: The last digit is $0$:

   • If the last digit is $0$, we need to select $2$ additional digits from $\{1, 2, 3, 4, 5, 6\}$ (6 digits in total).
   • The number of ways to choose $2$ digits is: $\binom{6}{2} = \frac{6 \cdot 5}{2} = 15$
   • For each selection of $2$ digits, there are $2! = 2$ ways to arrange the first two digits, so the total number of favorable outcomes for this case is: $15 \cdot 2 = 30$

2. Case 2: The last digit is $5$:

   • If the last digit is $5$, we need to select $2$ additional digits from $\{0, 1, 2, 3, 4, 6\}$ (6 digits in total).
   • The number of ways to choose $2$ digits is: $\binom{6}{2} = 15$
   • For each selection of $2$ digits, there are $2! = 2$ ways to arrange the first two digits, so the total number of favorable outcomes for this case is: $15 \cdot 2 = 30$

Thus, the total number of favorable outcomes is: $30 + 30 = 60$

Probability

The probability is given by: $P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{60}{210} = \frac{2}{7}$

Final Answer:

The probability that a three-digit number formed from these digits is divisible by $5$ is: $\boxed{\frac{2}{7}}$

Do you want details on any step? Here are 5 related questions you might explore:

1. How is the total number of combinations calculated in problems like this?
2. What happens to the probability if digits are drawn with replacement?
3. Can you calculate the probability for other divisors, such as $3$ or $2$?
4. How does arranging digits affect the total outcomes?
5. What is the role of permutations versus combinations in this problem?

Tip: Always consider divisibility rules carefully when working with problems involving probabilities and digits.