Evaluate Subscript n Baseline Upper C Subscript x Baseline p Superscript x Baseline left parenthesis 1 minus p right parenthesis Superscript n minus x for nequals6​, pequals0.3​, xequals2.

To evaluate the given binomial probability expression:

\[
^nC_x \cdot p^x \cdot (1 - p)^{n - x}
\]

where \( n = 6 \), \( p = 0.3 \), and \( x = 2 \), we can break it down step by step.

### Step 1: Calculate \( ^nC_x \) (the binomial coefficient)
The binomial coefficient is given by:

\[
^nC_x = \frac{n!}{x!(n - x)!}
\]

Substituting \( n = 6 \) and \( x = 2 \):

\[
^6C_2 = \frac{6!}{2!(6 - 2)!} = \frac{6!}{2!4!} = \frac{6 \times 5}{2 \times 1} = 15
\]

### Step 2: Calculate \( p^x \)
Now we compute \( p^x \):

\[
p^x = (0.3)^2 = 0.09
\]

### Step 3: Calculate \( (1 - p)^{n - x} \)
Now, we calculate \( (1 - p)^{n - x} \):

\[
(1 - p)^{n - x} = (1 - 0.3)^{6 - 2} = (0.7)^4 = 0.2401
\]

### Step 4: Multiply all terms
Now we multiply all the terms together:

\[
15 \cdot 0.09 \cdot 0.2401 = 0.324135
\]

### Final Answer:
The value of the expression is approximately:

\[
0.324
\]

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**Relative Questions:**

1. How do you derive the binomial coefficient in general for any \( n \) and \( x \)?
2. What is the significance of the \( p^x \) term in a binomial probability formula?
3. Why is \( (1 - p)^{n - x} \) important in binomial distributions?
4. How would this change if \( p = 0.5 \) instead of \( p = 0.3 \)?
5. What are some practical examples of binomial distributions?

**Tip:** The binomial coefficient \( ^nC_x \) represents the number of ways to choose \( x \) successes from \( n \) trials.

Evaluate Subscript n Baseline Upper C Subscript x Baseline p Superscript x Baseline left parenthesis 1 minus p right parenthesis Superscript n minus x for n=6, p=0.3, x=2.

Learn how to evaluate the binomial probability expression for n=6, p=0.3, and x=2. This problem involves calculating the binomial coefficient, applying exponentiation, and multiplying the results. Suitable for Grades 10-12.

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