Happy Valentines Day 2015

Calculus Level 4

The UP Math Club celebrates Valentine's Day by conducting an event, "The Compatibility Test 2015". Thousands of people join in this event every year making this one of the most-awaited event of each year. This year, out of the 7700 people who attended this event, the club chose 2015 men and 2015 women for the Compatibility Test. I am one of the chosen men and ??????? is one of the chosen women. The test is conducted as follows:

  • The club created a large box with 4030 balls in it. Each ball has a unique integer written on it and all of the 4030 integers written in the 4030 balls are consecutive. (No one except the club knows what integers are written in the balls)

  • Each person randomly chooses a ball from the box with their corresponding gender.

  • A man and a woman will now be considered compatible if the function \[f(x) = x^4 - kx^3 + ax^2 - bx + ab\] has a root on the interval \([0 , 1]\) where "\(a\)" is the number picked by the man, "\(b\)" is the number picked by the woman and "\(k\)" is the current year which is \(2015\).

Based from these, If I picked the number \(7\), let p be the probability that ??????? is compatible to me. (Disregard the order of picking of the balls). Find \(\lfloor10000p\rfloor\)

Please show your solutions.


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