Question

In a certain college class, 40 percent of students scored an A on the final exam. What is the probability of randomly selecting two out of six students from the class who scored an A? Use the formula nCkpkqn−k
to solve.(1 point)
Responses

2 percent
2 percent

4 percent
4 percent

31 percent
31 percent

14 percent

Answer 1

Explain the steps to construct an approximate polynomial expression to represent a sequence with constant 4th differences of 48.

Answer 2

To solve this problem, we need to find the probability of randomly selecting exactly 2 students who scored an A from a group of 6 students, where 40% of students scored an A.

Let's denote:
- \( n = 6 \): the total number of students
- \( k = 2 \): the number of students who scored an A
- \( p = 0.4 \): the probability of a student scoring an A
- \( q = 1 - p = 0.6 \): the probability of a student not scoring an A

We can use the formula for combinations and the binomial probability formula:

\[
P(X = k) = \binom{n}{k} p^k q^{n-k}
\]

First, we need to calculate the binomial coefficient \( \binom{6}{2} \):

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

Next, we apply the probabilities:

\[
p^k = (0.4)^2 = 0.16
\]
\[
q^{n-k} = (0.6)^{6-2} = (0.6)^4 = 0.1296
\]

Now we can plug these values into the probability formula:

\[
P(X = 2) = \binom{6}{2} p^2 q^{4} = 15 \times 0.16 \times 0.1296
\]

Calculating this step-by-step:

1. Calculate \( 15 \times 0.16 \):
\[
15 \times 0.16 = 2.4
\]

2. Calculate \( 2.4 \times 0.1296 \):
\[
2.4 \times 0.1296 \approx 0.310656
\]

Finally, express this in percentage:

\[
0.310656 \times 100 \approx 31.07\%
\]

Thus, the probability of randomly selecting exactly 2 students who scored an A is approximately 31%.

Therefore, the answer is:

**31 percent**

Answer 3

To construct an approximate polynomial expression that represents a sequence with constant 4th differences, follow these steps:

### Step 1: Understanding Constant Differences
When a sequence has constant \(k\)-th differences, it can be represented by a polynomial of degree \(k\). For example, if:

- The first differences are calculated (differences of consecutive terms),
- The second differences (differences of the first differences),
- The third differences (differences of the second differences),
- The fourth differences (differences of the third differences),

If the fourth differences are constant, the sequence can be expressed as a polynomial of degree \(4\).

### Step 2: Finding the Leading Coefficient
Given that the constant fourth difference is \(48\), we can determine the leading coefficient of the polynomial. The constant \(d\) of the \(n\)-th difference relates to the leading coefficient \(a_4\) of the polynomial \(P(n) = a_4 n^4 + a_3 n^3 + a_2 n^2 + a_1 n + a_0\) by the formula:

\[
d = 24a_k,
\]

where \(d\) is the constant \(k\)-th difference (here \(d = 48\)), and \(a_k\) is the leading coefficient of the polynomial of degree \(k\).

To find \(a_4\):

\[
48 = 24a_4 \implies a_4 = \frac{48}{24} = 2.
\]

### Step 3: Setting Up the Polynomial
Now we know the leading coefficient \(a_4 = 2\). The polynomial will have the form:

\[
P(n) = 2n^4 + a_3 n^3 + a_2 n^2 + a_1 n + a_0.
\]

### Step 4: Determine Remaining Coefficients
To determine \(a_3\), \(a_2\), \(a_1\), and \(a_0\), you would typically need additional information, such as initial terms of the sequence. However, if such values are not provided, you can assume a form or pattern for these coefficients based on the context or the specific behavior of the sequence.

### Example Coefficients Assignment
If we assume some arbitrary values for the coefficients based on a particular pattern or context, we might let:

- \(a_3 = 0\)
- \(a_2 = 0\)
- \(a_1 = 0\)
- \(a_0 = 0\)

In this case, the approximate polynomial expression representing the sequence is:

\[
P(n) = 2n^4.
\]

### Conclusion
In conclusion, to construct a polynomial expression for a sequence with constant 4th differences of \(48\):

1. Confirm that the polynomial degree is \(4\).
2. Use the constant difference to find the leading coefficient \(a_4\).
3. Construct the polynomial based on the leading coefficient and any available sequence values or assumptions for the lower-degree coefficients.

The final polynomial expression might look like \(P(n) = 2n^4 + C\), where \(C\) can be further refined with more specific information about the sequence terms.