Answer:
Check the explanation
Step-by-step explanation:
(a)Let p be the smallest prime divisor of (n!)^2+1 if p<=n then p|n! Hence p can not divide (n!)^2+1. Hence p>n
(b) (n!)^2=-1 mod p now by format theorem (n!)^(p-1)= 1 mod p ( as p doesn't divide (n!)^2)
Hence (-1)^(p-1)/2= 1 mod p hence [ as p-1/2 is an integer] and hence( p-1)/2 is even number hence p is of the form 4k+1
(C) now let p be the largest prime of the form 4k+1 consider x= (p!)^2+1 . Let q be the smallest prime dividing x . By the previous exercises q> p and q is also of the form 4k+1 hence contradiction. Hence P_1 is infinite
SOLUTION
This is a binomial probability. For i, we will apply the Binomial probability formula
i. Exactly 2 are defective
Using the formula, we have

Note that I made the probability of being defective as the probability of success = p
and probability of none defective as probability of failure = q
Exactly 2 are defective becomes the binomial probability

Hence the answer is 0.1157
(ii) None is defective becomes

hence the answer is 0.4823
(iii) All are defective

(iv) At least one is defective
This is 1 - probability that none is defective

Hence the answer is 0.5177
Equivalent expressions are expressions of equal values
The simplified expression of (3 - 4i) + (7 - 6i) is 
The expression is given as:

Remove brackets in the above expression

Rewrite the expression by collecting the like terms

Next, evaluate the like terms

Hence, the simplified expression of (3 - 4i) + (7 - 6i) is 
Read more about equivalent expressions at:
brainly.com/question/2972832
Answer:
The probability of eating pizza given that a new car is bought is 0.952
Step-by-step explanation:
This kind of problem can be solved using Baye’s theorem of conditional probability.
Let A be the event of eating pizza( same as buying pizza)
while B is the event of buying a new car
P(A) = 34% = 0.34
P(B) = 15% = 15/100 = 0.15
P(B|A) = 42% = 0.42
P(B|A) = P(BnA)/P(A)
0.42 = P(BnA)/0.34
P(B n A) = 0.34 * 0.42 = 0.1428
Now, we want to calculate P(A|B)
Mathematically;
P(A|B = P(A n B)/P(B)
Kindly know that P(A n B) = P(B n A) = 0.1428
So P(A|B) = 0.1428/0.15
P(A|B) = 0.952