https://file.aiquickdraw.com/mathbot/file/1737415427834g6mflbqe.jpg

The image contains a triangular arrangement of numbers, which corresponds to Pascal's Triangle. It is used for calculating combinations. The entries in Pascal's Triangle represent binomial coefficients.

We need to calculate the following combinations:

1. \( 3C2 \)
2. \( 3C3 \)

Also, the image mentions the second entry in the fourth row of Pascal's Triangle, which is 6, and \( C2 \).

### Let's break this down:

#### **Pascal's Triangle:**
In Pascal's Triangle:
- The first entry in each row corresponds to \( C(0, 0) = 1 \).
- Each entry in a row can be computed by adding the two numbers directly above it from the previous row.

#### **Combination formula:**
The combination formula is given by:
\[
nCk = \frac{n!}{k!(n-k)!}
\]
Where:
- \( n \) is the total number of items.
- \( k \) is the number of items to choose.

### To calculate:

#### 1. \( 3C2 \)
We have:
\[
3C2 = \frac{3!}{2!(3-2)!} = \frac{3!}{2!1!}
\]

#### 2. \( 3C3 \)
We have:
\[
3C3 = \frac{3!}{3!(3-3)!} = \frac{3!}{3!0!}
\]

Let me now compute these values for you. math

# Calculate combinations
nC2 = math.comb(3, 2)
nC3 = math.comb(3, 3)

nC2, nC3The calculated values for the combinations are:

1. \( 3C2 = 3 \)
2. \( 3C3 = 1 \)

Thus:
- \( 3C2 = 3 \)
- \( 3C3 = 1 \)

Would you like more detailed explanations on these calculations or help with another question? Here are some related questions:

1. What is the general pattern of numbers in Pascal's Triangle?
2. How can you calculate higher combinations, such as \( 5C3 \)?
3. What is the significance of Pascal's Triangle in binomial expansions?
4. How do you calculate combinations using factorials manually?
5. How are combinations different from permutations?

**Tip:** To efficiently calculate combinations, remember that using the formula \( \frac{n!}{k!(n-k)!} \) reduces repetitive factorial calculations.

Calculate the combinations: 3C2 and 3C3, and find the second entry in the fourth row of Pascal's Triangle.

Explore how to calculate the combinations 3C2 and 3C3 using Pascal's Triangle and the combination formula nCk = n! / [k!(n-k)!]. Understand the connection between Pascal's Triangle and binomial coefficients, with step-by-step explanations. Ideal for students in Grades 8-10.

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