Question

1. your locker has a combination lock. the combination consists of a specific 3-number code. the numbers go from 0 to 49 and they can repeat. how many codes are possible?
2. a smartphone lets you set a 4-digit passcode using the digits 0-9. digits can repeat. how many possible passcodes exist?
3. a security code is made up of 3 digits followed by 4 letters. digits can repeat, but letters cannot. how many codes are possible?
4. a video game allows you to create a character by choosing a species (5 options), a weapon (8 options), and a skill set (6 options). how many unique characters are possible?

Explanation:

Let's solve question 8 first (assuming it's "A smartphone lets you set a 4 - digit passcode using the digits 0 - 9. Digits can repeat. How many possible passcodes exist?"):

Step1: Analyze each digit position

For a 4 - digit passcode, each of the 4 positions (thousands, hundreds, tens, ones) can be filled with a digit from 0 - 9.

Step2: Calculate the number of choices for each position

Since digits can repeat, for each of the 4 positions, there are 10 possible choices (the digits 0, 1, 2, ..., 9).

Step3: Use the multiplication principle

The total number of possible passcodes is the product of the number of choices for each position. So we calculate $10\times10\times10\times10$.
We know that $10\times10\times10\times10 = 10^{4}=10000$.

Now question 9: "A security code is made up of 3 digits followed by 4 letters. Digits can repeat, but letters cannot. How many codes are possible?"

Step1: Calculate the number of ways to choose the 3 digits

For each of the 3 digit positions, there are 10 possible digits (0 - 9) since digits can repeat. By the multiplication principle, the number of ways to choose the 3 digits is $10\times10\times10=10^{3} = 1000$.

Step2: Calculate the number of ways to choose the 4 letters

There are 26 letters in the alphabet. For the first letter position, we have 26 choices. Since letters cannot repeat, for the second letter position, we have 25 choices (because we already used one letter), for the third letter position, we have 24 choices, and for the fourth letter position, we have 23 choices. The number of ways to choose the 4 non - repeating letters is $26\times25\times24\times23=\frac{26!}{(26 - 4)!}=26\times25\times24\times23 = 358800$.

Step3: Calculate the total number of security codes

To find the total number of security codes, we multiply the number of ways to choose the digits by the number of ways to choose the letters. So we have $1000\times358800 = 358800000$? Wait, no, wait: Wait, 2625 = 650, 2423=552, 650552 = 650500+65052=325000 + 33800=358800. Then 101010=1000. Then 1000358800 = 358800000? Wait, maybe I made a mistake. Wait, no, the problem says 3 digits followed by 4 letters. Wait, 3 digits: each digit 0 - 9, repeat allowed: 10^3=1000. 4 letters: no repetition. The number of permutations of 26 letters taken 4 at a time is $P(26,4)=\frac{26!}{(26 - 4)!}=26\times25\times24\times23 = 358800$. Then total number of codes is $10^{3}\times P(26,4)=1000\times358800 = 358800000$? Wait, but maybe the letters are case - insensitive? If we assume only lowercase (or only uppercase), the above is correct. But maybe the original problem has a typo or I misread. Wait, maybe the letters can be in any case? But usually, if not specified, we assume 26 letters. Alternatively, maybe the problem is 3 digits and 4 letters with letters repeating? But the problem says "letters cannot". So according to the problem statement, the calculation is $10\times10\times10\times26\times25\times24\times23=1000\times358800 = 358800000$. But maybe I made a mistake in the problem reading. Let's re - check: "3 digits followed by 4 letters. Digits can repeat, but letters cannot." So digits: 3 positions, 10 choices each: 10^3. Letters: 4 positions, 26,25,24,23 choices: 26252423. Multiply them: 1000358800 = 358800000.

Question 10: "A video game allows you to create a character by choosing a species (5 options), a weapon (8 options), and a skill set (6 options). How many unique characters are possible?"

Step1: Identify the number of choices for each category

We have 3 categories: species (5 options), weapon (8 options), and skill set (6 options).

Step2: Apply the multiplication principle

To find the total number of unique characters, we multiply the number of choices for each category. So we calculate $5\times8\times6$.
First, $5\times8 = 40$, then $40\times6=240$.

For question 7 (the first one, "Your teacher has a combination lock. The combination consists of a specific 3 - number code. The numbers must be 0 - 9 and they can repeat. How many codes are possible?"):

Answer:

10000