i am going to install 5 new game apps on my phone. i shortlisted 2 word games, 5 quizzes and 16 saga games. i want at least one of each type of game. how many different selections of apps could i possibly choose from

To solve this problem, we need to calculate how many different ways you can select 5 game apps from the given list, with the condition that you must select at least one app from each category (word games, quizzes, and saga games).

You shortlisted:

- 2 word games
- 5 quizzes
- 16 saga games

### Step 1: Choose 1 game from each category
Since you must choose at least one game from each type, let’s first select:

- **1 word game** from 2 available: This can be done in \( \binom{2}{1} = 2 \) ways.
- **1 quiz game** from 5 available: This can be done in \( \binom{5}{1} = 5 \) ways.
- **1 saga game** from 16 available: This can be done in \( \binom{16}{1} = 16 \) ways.

### Step 2: Select the remaining 2 apps
Now, you have already chosen 3 games (1 from each category), and you need to choose 2 more from the remaining games. Since you can choose from any type of game now, there are:

- 1 word game left (since you already picked 1 out of 2)
- 4 quizzes left (since you picked 1 out of 5)
- 15 saga games left (since you picked 1 out of 16)

So there are \( 1 + 4 + 15 = 20 \) games remaining. You need to choose 2 games from these remaining 20 games. The number of ways to do this is \( \binom{20}{2} \).

The number of ways to choose 2 games from 20 is:

\[
\binom{20}{2} = \frac{20 \times 19}{2} = 190
\]

### Step 3: Total number of selections
Now, multiply the number of ways to choose 1 game from each category by the number of ways to choose the remaining 2 games:

\[
\text{Total ways} = 2 \times 5 \times 16 \times 190
\]

\[
\text{Total ways} = 2 \times 5 \times 16 \times 190 = 30400
\]

So, there are **30,400** different possible selections of 5 apps, with at least one game from each type.

Would you like further clarification or more details?

Here are some related questions to think about:
1. What if you were required to choose exactly 2 word games?
2. How would the number of selections change if you could select all 5 games from saga games only?
3. What happens if you had to choose at least 2 games from quizzes instead of just 1?
4. How many selections are possible if you add one more word game to the shortlist?
5. What would be the result if you reduced the number of quizzes by 2?

**Tip**: When making selections with conditions, always consider how restrictions impact the total choices!

Find out how to calculate the different selections of 5 game apps from a list of 2 word games, 5 quizzes, and 16 saga games, ensuring at least one from each category. Learn combinatorics concepts like selections with restrictions and use the combination formula. Suitable for high school students.

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