Hold on to your hat - this is a fun problem that might get a bit complicated...

A school has two houses, each with 50 students. The sorting hat places students in the two houses somewhat at random, but based on whether they have the following traits:

• Athletic
• Brave
• Creative

In House 1:

• All the students are Athletic
• 3x students are not Creative
• The number of students that are both Brave and Creative is twice the number of students that are neither
• 90% of the students are either Brave, or they are Creative but not Brave

In House 2:

• 30% of the students possess no traits
• 5x students possess at least 2 traits
• The number of students who possess all 3 traits is quadruple the number of students who possess exactly 2 traits
• 15 students are Creative but not Brave

(1) Find the possible values of x.

(2) In each part, a new rule is added to the above information. For each part, find the new possible values of x. (The new rule only applies for that part of the question.)

(a) 40% of students in House 2 possess exactly one trait

(b) The probability that a randomly selected student is Creative is 67%

(c) A randomly selected Athletic student has a probability of 24/74 of being in House 2.

(3) What is the least common trait overall?

(4) Prove that it is impossible to have 39 Brave students in total. Are there any other impossible numbers of Brave students?

 

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