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Question: In how many different orders can Debbie, Brigitte, and Adam wait in line in the school cafeteria? Wh...

In how many different orders can Debbie, Brigitte, and Adam wait in line in the school cafeteria? What is the probability that they will be in alphabetical order?

Explanation

Solution

To solve this question, we need to find the different number ways in which the 3 people can be arranged. This can be done by counting the number of possibilities to arrange Debbie, Brigitte and Adam. We then need to consider the number of cases in which they will be arranged in alphabetical order and find the probability of this happening in the total number of possibilities.

Complete step by step answer:
According to the given question, we have to find the different orders in which the 3 people Debbie, Brigitte, and Adam wait in line in the school cafeteria. This can be calculated as follows:
There are 3 positions now to be occupied by the 3 students. The first position can be occupied by any of the 3 students. After we fill this position with a student, we have 2 positions remaining. We can fill this position with any 1 of the remaining 2 students. Now we have two positions occupied and only 1 remaining to be filled by the remaining student. This can be given mathematically as,
⇒3×2×1 ⇒3!=6 \begin{aligned} & \Rightarrow 3\times 2\times 1 \\\ & \Rightarrow 3!=6 \\\ \end{aligned}
Therefore, there are 6 ways in which we can arrange the 3 students in 3 positions.
Now we need to find out the probability of arranging them in alphabetical order and there is only one such case: Adam first, Brigitte second and Debbie third. So out of the possible 6 cases, there is only 1 such case where they will be in the line in alphabetical order.
Therefore, the probability for this is given as
⇒P(in the line in alphabetical order)=16\Rightarrow P\text{(in the line in alphabetical order)}=\dfrac{1}{6}

Hence, there are 6 different ways in which Debbie, Brigitte, and Adam can wait in line in the school cafeteria and the probability that they will be in alphabetical order is 16.\dfrac{1}{6}.

Note: This question can be solved by drawing a table for the students to be arranged in different orders considering each case one by one. This is easy if the number of students is small. However, it becomes tedious to solve for a large number of students. In that case we use the general formula for arranging n students in n places which is n!.n!.