Question

Imagine, just imagine, I am in the mood to give away 7 extra credit points among my (only) 4 students. If all students have the same opportunity to get any of the points, what is the probability that each student gets at least one extra point?

Answer 1

Here let say for each student get x₁ ,x₂ ,x₃ and x₄ credit points, so here

x₁ + x₂ + x₃ + x₄ = 7

where

x₁ ,x₂ ,x₃ ,x₄ >= 0

So total number of possibilities here are as per combinamatrics formula = ^{(7 + 4 -1)}C_(4-1) = ¹⁰C₃ = 120

Now, we know that each of the student get atleast one extra point so here there is only three extra point are left for distribution, so if y₁,y₂ ,y₃ and y₄ are the extra credit received by the students.

So total number of possibilities here are as per combinamatrics formula = ^{(3 + 4 -1)}C_(4-1) = ⁶C₃ = 20

so here probability that atleast each student gets at least one extra point = 20/120 = 1/6