Combinatorics | Number Theory Answers

i a grup of students some are i oe or more of scool hockey,netball or swimmig teams.the diagram show
the numbers where h are i hockey team,n are in netball team nd sare i swimmig team.if tere are 21 pupil find x? what is the probabilty that pupil chose at random is i 1. hockey team only 2.at least two teams 3. exactly one team 4.the net ball team 4 3 3
2 2 1

1 x=? these numbers are in 3 circles

This question is with regards to Fermat Last Theorem and Number Theory.

Let p and θ be primes with θ>p such that p is not a factor of θ−1. As θ is a prime, we know that for any x ∈ Z we can write xθ−1 in the form xθ−1 = kθ+1 for some integer k. Furthermore, since the greatest common divisor of p and θ − 1 is 1, we can write 1 = ap + b(θ − 1) for some integers a and b.
(a) Show that every integer x can be written in the form x = yp + jθ for some integer y and some integer j.
(b) Show that for p a prime, any prime φ that satisfies the conditions of Lemma 4.1 has the property that p is a factor of φ − 1.
(c) Prove the First Case of Fermat’s Last Theorem for the exponents 13, 17 and 19. (That is, show that if there is a nonzero integer solution to xp + yp = zp for p = 13 then 13 is a factor of xyz, and so on.)

Let us define the height h(a) of an integer a >= 2 to be the greatest n such that Euclid's algorithm requires n steps to compute gcd(a, b) for some positive b < a. Show that h(a) = 1 if and only if a = 2, and find h(a) for all a <= 8.

if x be normal with mean 10 and variance 4 ,then P(x<11)==