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Combinatorics is a field of mathematics concerned with counting, arrangement, and combination of objects. In our context, combinatorics helps us determine the number of ways a rumor can circulate among a group of friends. It allows us to set up the problem by understanding how choices among a particular set can be made.

• Combinatorics covers various concepts, such as permutations and combinations.
• Permutations focus on the arrangement of objects in a specific order, which is precisely what we're looking at when determining the order the friends hear the rumor.

For example, in the problem, if we wanted to find the total number of ways to arrange the order in which five friends might hear a rumor, we would calculate this using factorial notation. Specifically, with five friends, the permutations would be represented by: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]This equation tells us that there are 120 different ways to arrange the sequence in which the rumor can be heard among friends B, C, D, E, and F.

Permutations involve arranging items in a specific sequence or order. The key feature of permutations is that the order matters. In the given problem, every permutation signifies a unique sequence in which the rumor is passed.

• The problem requires understanding that each choice by a person affects the subsequent choices and overall sequence.
• It's important to realize that with each person hearing the rumor, there are progressively fewer people left to choose from.

For the step-by-step solution to part a, for example, we calculated the probability of the specific order B, C, D, E, and F by considering these selections:

• The first choice has 5 options: \(\frac{1}{5}\)
• The next choice out of four remaining: \(\frac{1}{4}\)
• Then out of three: \(\frac{1}{3}\)
• Out of two, and finally one: \(\frac{1}{2}\) and last as \(1\)

By multiplying these probabilities, we find that the probability of the exact sequence B, C, D, E, F occurring is \(\frac{1}{120}\). This reflects how permutations require specific ordered arrangements.

Conditional probability is the likelihood of an event happening given that another event has already occurred. In our exercise, we found conditional probabilities to be key when determining probabilities of different rumor sequences.

• It first appears in part b, where we seek the probability that F is the third to hear the rumor given certain conditions are met.
• Here, conditional probability helps understand how preceding events affect subsequent probabilities.

For part b, the event that F is the third listener is conditional on the choices of the first two people. We calculated it as: \[ \frac{4}{5} \times \frac{1}{4} \times \frac{3!}{6} = \frac{2}{15} \]This entails multiplying the probability of the first two selections and considering all possible permutations of the remaining people. Similarly, part c involves F being last, which depends on none of the first four people picking F. Through conditional probability, the intricate nature of how earlier choices condition following probabilities becomes clear, especially where the order and specific roles of friends in sequence matter.