Question

part (e) you are to find the probability of the five die showing two distinct pairs. select the probability expression that would correctly compute this probability.
a. (\frac{\binom{6}{2}\binom{4}{1}\binom{5}{2}\binom{3}{2}}{\binom{6}{5}})
b. (\frac{\binom{6}{2}\binom{4}{1}\binom{5}{2}\binom{3}{2}}{6^{5}})
c. (\frac{\binom{6}{2}\binom{4}{1}\binom{5}{2}\binom{3}{2}}{\binom{6}{5}})
d. (\frac{\binom{6}{2}}{6^{5}})
e. (\frac{\binom{6}{2}\binom{4}{1}\binom{5}{2}\binom{3}{2}}{6^{5}})

Explanation:

Step1: Calculate total number of outcomes

When rolling 5 - die, each die has 6 possible outcomes. So the total number of possible outcomes is $6^5$.

Step2: Calculate number of favorable outcomes

1. Choose 2 distinct numbers for the pairs: The number of ways to choose 2 distinct numbers out of 6 for the pairs is $C(6,2)=\frac{6!}{2!(6 - 2)!}=\frac{6\times5}{2\times1}=15$.
2. Choose 2 positions for the first - pair and 2 positions for the second - pair out of 5 positions: The number of ways to choose 2 positions for the first pair out of 5 positions is $C(5,2)=\frac{5!}{2!(5 - 2)!}=10$, and then the number of ways to choose 2 positions for the second pair out of the remaining 3 positions is $C(3,2)=\frac{3!}{2!(3 - 2)!}=3$. The remaining 1 position will have a non - paired number. After choosing 2 numbers for the pairs, there are 4 remaining numbers for the non - paired position.
3. Calculate the number of favorable outcomes: The number of favorable outcomes is $C(6,2)\times C(5,2)\times C(3,2)\times4=\frac{6!}{2!4!}\times\frac{5!}{2!3!}\times\frac{3!}{2!1!}\times4$.
4. Probability formula: The probability $P$ of an event is the number of favorable outcomes divided by the total number of outcomes. So the probability of getting two distinct pairs when rolling 5 die is $\frac{C(6,2)\times C(5,2)\times C(3,2)\times4}{6^5}$.

Answer:

The correct probability expression is the one that follows the above - derived logic. Without seeing the actual expressions with proper binomial and factorial notations in the options (since the provided options are not clear in the image), the general form of the correct probability expression for the event of getting two distinct pairs when rolling 5 die is $\frac{C(6,2)\times C(5,2)\times C(3,2)\times4}{6^5}$. If we assume the options are written in terms of binomial coefficients $C(n,k)=\binom{n}{k}$, we need to match the structure of the formula we derived.