Let's solve the problem step-by-step to find which expression represents the probability of rolling a 5 exactly three times in ten rolls of a number cube with six sides.
### Step 1: Understand the Binomial Probability Formula
When dealing with a binomial probability (like the probability of a specific outcome across several trials), we use the formula:
[tex]\[ P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k} \][/tex]
Where:
- [tex]\( n \)[/tex] is the total number of trials (rolls of the number cube).
- [tex]\( k \)[/tex] is the number of successes (number of times a 5 is rolled).
- [tex]\( p \)[/tex] is the probability of success on a single trial.
- [tex]\( \binom{n}{k} \)[/tex] is the combination function, representing the number of ways to choose [tex]\( k \)[/tex] successes from [tex]\( n \)[/tex] trials.
### Step 2: Identify the Values
For this specific problem:
- [tex]\( n = 10 \)[/tex] (total of 10 rolls).
- [tex]\( k = 3 \)[/tex] (we need exactly 3 rolls where we get a 5).
- [tex]\( p = \frac{1}{6} \)[/tex] (probability of rolling a 5 since a number cube has six sides).
### Step 3: Write the Binomial Probability Expression
Now substitute these values into the binomial probability formula:
[tex]\[ P(X = 3) = \binom{10}{3} \cdot \left(\frac{1}{6}\right)^3 \cdot \left(\frac{5}{6}\right)^{10-3} \][/tex]
### Step 4: Identify the Correct Expression from the Choices
Compare the calculated expression to the given choices:
[tex]\[
\begin{array}{l}
10^{c_3} \left( \frac{1}{6} \right)^3 \left( \frac{1}{6} \right)^7 \\
10^{c_3} \left( \frac{1}{2} \right)^3 \left( \frac{1}{2} \right)^7 \\
10^{c_3} \left( \frac{1}{6} \right)^3 \left( \frac{5}{6} \right)^7 \\
10^{c_3} \left( \frac{1}{6} \right)^7 \left( \frac{5}{6} \right)^3
\end{array}
\][/tex]
From these, the correct matching expression is:
[tex]\[ 10^{c_3} \left( \frac{1}{6} \right)^3 \left( \frac{5}{6} \right)^7 \][/tex]
### Conclusion
The correct expression for representing the probability of rolling a 5 exactly three times in ten rolls of a number cube with six sides is:
[tex]\[ 10^{c_3} \left( \frac{1}{6} \right)^3 \left( \frac{5}{6} \right)^7 \][/tex]
