Let n = 2k with k ≥ 2. (a) Prove that pk is an element of order 2 which commutes with all elements of D2n. (b) Prove that pk is the only non-identity element which commutes with all elements of D2n.

A company produces scooters used by small businesses that find them convenient for making short deliveries. The company is notified whenever a scooter breaks down, and the problem is classified as being either mechanical or electrical. The company then matches the scooter to the plant where it was assembled. The data table to the right contains a random sample of 200 breakdowns. Use the data file and the relative frequency assessment method to complete parts a through c. a. What is the probability a scooter was assembled at Plant 2? P(Plant 2) = (Type an integer or a decimal.) b. What is the probability that a scooter breakdown was due to an electrical problem? P(electrical) = (Type an integer or a decimal.) C. What is the probability that a scooter breakdown was due to an electrical problem and the scooter was assembled at Plant 1? P(electrical and Plant 1) = (Type an integer or a decimal.)

The number of digits in the CVC was changed from 3 digits to 4 digits. Determine the increase in number ofpossible CVCs .

A) A class of 2018 graduates  were 1250 students, 40% of them were female students and the rest were Male, 10% of the female students  got a grade of A and 10 students of the female grades A were international students.  Class of 2019 had 875 Male students,  also 10 female students  got an A in their grade in class 2019, where the total number of students  did not change from last year. Calculate the following. Show your calculations step by step.  The number of female students. The number of Female students with a grade A  The percentage of the female international students graded A out of the total number of  female students with a grade A. What is the percentage increase  or decrease in the number of Male students in comparison to class 2018? What is the percentage of female students  who got A out of the female students? B) 23, 30, 1,49, 2, 36, 7, 23, 13,9, 30, 17, 20. Calculate the mean, median and mode.