Answer:
The probability the he or she will answer correctly is 1.5%
Step-by-step explanation:
In all, there are 18 problems. In this question, the order of which the problems are sorted for the quiz makes no difference. For example, if the question A of the quiz is P1 and question B P2, and question A P2 and question B P1, it is the same thing.
There are 18 problems and 9 are going to be selected. So, there is going to be a combination of 9 elements from a set of 18 elements.
A combination of n elements from a set of m objects has the following formula:
![C_{(m,n)} = \frac{m!}{n!(m-n)!}](https://tex.z-dn.net/?f=C_%7B%28m%2Cn%29%7D%20%3D%20%5Cfrac%7Bm%21%7D%7Bn%21%28m-n%29%21%7D)
In this question, m = 18, n = 9. So the total number of possibilities is:
![T_{p} = C_{(18,9)} = \frac{18!}{9!(18-9)!} = 48620](https://tex.z-dn.net/?f=T_%7Bp%7D%20%3D%20C_%7B%2818%2C9%29%7D%20%3D%20%5Cfrac%7B18%21%7D%7B9%21%2818-9%29%21%7D%20%3D%2048620)
Now we have to calculate the number of desired outcomes. This number is a combination of 9 elements from a set of 13 elements(13 is the number of problems that the student has figured out how to do).
Now, m = 13, n = 9. The number of desired possibilities is:
![D_{p} = C_{(13,9)} = \frac{13!}{9!(13-9)!} = 715](https://tex.z-dn.net/?f=D_%7Bp%7D%20%3D%20C_%7B%2813%2C9%29%7D%20%3D%20%5Cfrac%7B13%21%7D%7B9%21%2813-9%29%21%7D%20%3D%20715)
The probability is the number of desired possibilities divided by the number of total possibilities. So
![P = \frac{715}{48620} = 0.015 = 1.5%](https://tex.z-dn.net/?f=P%20%3D%20%5Cfrac%7B715%7D%7B48620%7D%20%3D%200.015%20%3D%201.5%25)
The probability the he or she will answer correctly is 1.5%