Liz is a student in Ms. Xu's class. Liz says to her classmates, "Of all the pairs of students Ms. Xu can choose as class leaders, I am included in one-tenth of those pairs." How many students are in Ms. Xu's class?
Assuming that all of the students in the class could be selected without bias.
The count of ways to select any 2 students from a class of size n, is:
{n!}/{2!*(n-2)!}
The count of ways to select Liz and any one other student from the class is:
(n-1)
If Liz's statement is accurate then:
1/10 = {(n-1)*2!*(n-2)!}/{n!}
Simplify and solve for n, the count of students in the class.
To find out how many students are in Ms. Xu's class, we need to set up an equation based on the information given.
Let's say there are "n" students in the class. The number of possible pairs of students that Ms. Xu can choose as class leaders would be given by the formula n choose 2, denoted as nC2.
According to Liz's statement, she is included in one-tenth of those pairs. So, we can assume that Liz is present in nC2/10 pairs.
Now we can set up the equation:
nC2/10 = 1
Simplifying the equation, we get:
n(n-1)/2 * 1/10 = 1/1
Now we can solve for "n":
n(n-1)/20 = 1
Multiplying both sides of the equation by 20 gives:
n(n-1) = 20
Expanding the equation, we get:
n^2 - n = 20
Rearranging the equation, we get:
n^2 - n - 20 = 0
Now we need to find the values of "n" that satisfy this quadratic equation. We can factorize the equation:
(n-5)(n+4) = 0
So, the possible solutions are n = 5 or n = -4. Since the number of students cannot be negative, we discard the -4.
Therefore, the number of students in Ms. Xu's class is 5.