Question

A total of $n$students are enrolled in a review course for the actuarial examination in probability. The posted

results of the examination will list the names of those who passed, in decreasing order of their scores. For instance, the posted result will be “Brown, Cho” if Brown and Cho are the only ones to pass, with Brown receiving the higher score. Assuming that all scores are distinct (no ties), how many posted results are possible?

Short answer

The possible no. of posted results are $\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)k!$

Step-by-step solution

Step 1. Given information.

It is given that,

Total no. of students enrolled in a review course for the actuarial examination $=n$.

The posted results of the examination will list the names of those who passed, in decreasing order of their scores.

Step 2. Find the possible no. of posted results.

If $k$people pass then there are $\left(\begin{array}{c}n\\ k\end{array}\right)$different groups of size $k$.

Further there are $k!$possible ordering of their scores.

Therefore, there are $\left(\begin{array}{c}n\\ k\end{array}\right)k!$possible results in which $k$people pass.

The possible no. of posted results are$\sum _{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)k!$.