https://file.aiquickdraw.com/mathbot/file/1733930587566vznftg2z.jpg

Let us break this problem into steps and solve each part systematically:

### Given Information:
- **6 black balls**, **2 white balls**, and **4 red balls**.
- **Total balls** = \(6 + 2 + 4 = 12\).

We are drawing **3 balls simultaneously**.

---

### a) **How many total draws are possible?**

The total number of ways to choose 3 balls from 12 is given by the combination formula:
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

Here \(n = 12\) and \(k = 3\):
\[
\binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220
\]

Thus, **220 total draws are possible**.

---

### b) **How many draws consist of balls of three different colors?**

To have balls of three different colors, we must select:
- 1 black ball,
- 1 white ball,
- 1 red ball.

The number of ways to select 1 ball of each color is:
\[
\binom{6}{1} \times \binom{2}{1} \times \binom{4}{1}
\]

Calculating each term:
\[
\binom{6}{1} = 6, \quad \binom{2}{1} = 2, \quad \binom{4}{1} = 4
\]

So:
\[
6 \times 2 \times 4 = 48
\]

Thus, **48 draws consist of balls of three different colors**.

---

### c) **How many draws consist of balls of the same color?**

To have all 3 balls of the same color, they must all be:
1. Black balls, or
2. White balls, or
3. Red balls.

The number of ways to choose 3 balls of the same color is:
- For black balls: \(\binom{6}{3}\),
- For white balls: \(\binom{2}{3}\) (not possible as there are only 2 white balls),
- For red balls: \(\binom{4}{3}\).

Calculating each term:
\[
\binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20, \quad \binom{2}{3} = 0, \quad \binom{4}{3} = \frac{4 \times 3 \times 2}{3 \times 2 \times 1} = 4
\]

Adding these:
\[
20 + 0 + 4 = 24
\]

Thus, **24 draws consist of balls of the same color**.

---

### Final Answers:
a) Total draws possible: **220**  
b) Draws with three different colors: **48**  
c) Draws with balls of the same color: **24**

Would you like a detailed explanation of any step or further assistance?  

---

#### 5 Related Questions:
1. How would the results change if we drew 4 balls instead of 3?
2. What is the probability of getting exactly two red balls in a draw of 3?
3. How many ways can you draw 2 balls of the same color and 1 of a different color?
4. What is the probability of drawing no black balls in a single draw of 3?
5. How many different distributions are possible if 4 balls are drawn at once?

#### Tip:
When solving combinatorics problems, always verify whether repetition is allowed (permutations vs. combinations) to avoid calculation errors.

Un sac contient six boules noires, deux boules blanches et quatre boules rouges. On tire au hasard simultanément trois boules du sac. Combien y a-t-il de tirages possibles ? Combien de tirages sont constitués de trois boules de couleurs différentes ? Combien de tirages sont constitués de trois boules de la même couleur ?

Solve a combinatorics problem involving drawing 3 balls from a bag containing 6 black balls, 2 white balls, and 4 red balls. Learn how to calculate total possible draws, draws with balls of different colors, and draws with balls of the same color. Suitable for Grades 9-12.

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