a) How many different initials can someone have if a person has at least two, but no more than five, initials? You may assume that each initial is one of the 26 uppercase letters of English, and that letters can be repeated.
b) When attempting to name his son, Jor El has decided on the following rules. i) The name must have exactly five letters ii) the second and fourth letters must be vowels and iii) The third and fifth letters must be the same. Assuming that he can choose from the 26 letters in the English alphabet, how many different names can Jor El pick for his son?
c) A professor designed his final exam as follows: There will be three sections in the exam. Each section has 5 questions. Students have to answer 4 questions from A, 3 from B and only 2 from C. It does not matter in what order the students answer the questions within a section. In how many possible ways can a student choose which questions to answer?
d) A professor designed his final exam as follows: There will be three sections in the exam. Each section has five questions. Students have to pick any two sections to answer, in any order. Within each section, they must choose any three questions. In how many possible ways can a student choose which questions to answer?
e) Computer passwords are to consist of a string of six symbols taken from the digits 0-9 and the lowercase letters. How many computer passwords have a repeated symbol?
f) How many odd numbers between 10000 and 99999 have distinct digits?
g) A classroom has two rows of eight seats each. There are 14 students, 5 of whom always sit in the front row and 4 of whom always sit in the back row. In how many ways can the students be seated?
h) A committee of five people is to be chosen from a club that has 10 male members and 12 female members. How many ways can the committee be formed if it is to contain at least two women?
i) A committee of five people is to be chosen from a club that has ten scientists and eight engineers. How many ways can the committee be formed if it has to contain at least two scientists and at least one engineer?
j) There are 10 men and 7 women working as supervisors in a company. The company has recently decided to form a committee to represent all the employees. The committee has to consist of 3 members, all of whom must be supervisors. The members will be President, General Secretary and Coordinator respectively. Answer the following questions based on this information.
i) How many ways can the committee be formed from the supervisors available?
ii) How many ways can the committee be formed if the General Secretary must be a female?
iii) How many ways can the committee be formed if it must have at least one man and at least one woman?
The Attempt at a Solution
a) 262 + 263 + 264 + 265
e) (36)8 - (36P8)
g) (8P8)(8P6) + (8P7)(8P7) + (8P6)(8P8)
h) (12C2)(10C3) + (12C3)(10C2) + (12C4)(10C1) + (12C5)(10C0)
i) (10C2)(8C3) + (10C3)(8C2) + (10C4)(8C1)
j) i) 17C3 ii) (7C1)(16C2) iii) (10C1)(7C2) + (10C2)(7C1)
It would be nice if you could check the answers and point out any mistakes.