A randomly generated password contains four characters. Each of the four characters is either a lowercase letter or a digit from 0-9. Each character in the password cannot be used more than once. What is the approximate probability that exactly one of the four characters will be a number? $$ 1 \% $$ $$ 11 \% $$ $$ 28 \% $$ $$ 44 \% $$

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To find the probability that exactly one of the four characters in the password is a number, we can follow these steps:

1. **Determine the total number of characters available**: 
   - There are 26 lowercase letters (a-z) and 10 digits (0-9), giving us a total of $$ 26 + 10 = 36 $$ characters.

2. **Calculate the total number of possible passwords**:
   - Since each character cannot be used more than once, the total number of ways to choose 4 characters from 36 is given by the permutation formula $$ P(n, r) = \frac{n!}{(n-r)!} $$.
   - Thus, the total number of passwords is $$ P(36, 4) $$.

3. **Calculate the number of favorable outcomes for exactly one digit**:
   - We need to choose 1 digit from the 10 available digits and 3 letters from the 26 available letters.
   - The number of ways to choose 1 digit from 10 is $$ \binom{10}{1} $$.
   - The number of ways to choose 3 letters from 26 is $$ \binom{26}{3} $$.
   - After choosing the characters, we can arrange them in $$ 4! $$ ways since the order matters.

4. **Calculate the probability**:
   - The probability is given by the ratio of the number of favorable outcomes to the total number of outcomes.

Now, let's perform the calculations step by step.

### Step 1: Total number of passwords
$$
P(36, 4) = \frac{36!}{(36-4)!} = \frac{36 \times 35 \times 34 \times 33}{1} = 1413720
$$

### Step 2: Number of favorable outcomes for exactly one digit
- Choosing 1 digit from 10:
$$
\binom{10}{1} = 10
$$
- Choosing 3 letters from 26:
$$
\binom{26}{3} = \frac{26 \times 25 \times 24}{3 \times 2 \times 1} = 2600
$$
- Arranging the 4 characters:
$$
4! = 24
$$

### Step 3: Total favorable outcomes
$$
\text{Favorable outcomes} = 10 \times 2600 \times 24 = 624000
$$

### Step 4: Calculate the probability
$$
P(\text{exactly 1 digit}) = \frac{624000}{1413720}
$$

Now, let's calculate this probability.

$$
P(\text{exactly 1 digit}) \approx 0.441
$$

To express this as a percentage:
$$
P(\text{exactly 1 digit}) \approx 44.1\%
$$

Thus, the approximate probability that exactly one of the four characters will be a number is $$ 44\% $$.

The answer is $$ 44\% $$.
The probability is approximately $$ 44\% $$.

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