Given the following problem :
A group of students got homework assignment where they need to pick a group of
functions and sort them .The students need to pick 5 pairs of functions and prove
some relation between them.
1.Given that there are 16 function in the group , how many pairs can we have ?
Answer : Using C(n,k) , we have C(16,2) = 16 above 2 = 120 , hence , we have 120 options for pairs .
2.Assuming that each student chose randomly the 5 pairs independently , what's the
probability that two students chose the exact same 5 pairs of functions ?
Answer : probability for choosing 5 function is : $5/120$ , where $120$ was calculated
in the previous exercise . Hence , the probability for two students to choose the exact
same functions is : $$(5/120)^2$$ .
3.Assume that we have 100 students , find the upper bound to the probability that there
are two students with the exact same 5 functions
Answer : $$C(100,2)∙(5/120)^2$$ = $$4950 ∙ (5/120)^2 $$
I'd appreciate any feedback and/or corrections regarding this exercise
Regards
EDIT :
Trying to fix part (3) :
Answer : $${\binom{100}2} ∙ \frac1{\binom{120}5}$$