Math Statistics Help

Question

I had some difficulty solving these questions. Id appreciate any help

A group of 30 people were asked if they were university graduates and if they belonged to a union. 16 were not graduates, 19 were union members, and 4 were non-union graduates. How many people were neither graduates nor union members?

(a) A security code consists of 5 uppercase letters. How many security codes are possible if we allow repetition?

(b) A security code consists of 5 uppercase letters. How many security codes are possible if we do not allow repetition?

(c) 9 switches are in a row on a computer terminal. Each switch is either on or off. How many different possible switch settings are there?

A library has 3 books from each of 3 different authors. How many ways can the library arange the books on the shelf if books by the same author must be placed together?

Solve each of the following: (a) P(7,6) (b) C(6,2) (c) P(7,7) (d) C(7,0)

Answer 1

30 people, with 19 union members, means 30 - 19 = 11 non-union members

11 non-union members - 4 non-union graduates = 7 non-union non-graduates

a)26 upper case letters; repetition allowed, 5 choices; 26^5 = 11881376

b)no repetition; order matters, so permutation; P(26,5) = 26*25*24*23*22 = 7893600

c) 9 switches, each on/off (2 choices) - 2^9 = 512

d)3 books, each of 3 authors, books together; For each author, 3! ways to arrange books; 3! ways to order authors

(3!)^3 * 3! = 6^4 = 1296

P(7,7) = 7*6*5*4*3*2*1 = 5040

P(7,6) = 7*6*5*4*3*2 = 5040

C(6,2) = 6*5/(2*1) = 15

C(7,0) = 1

Also, combinations and permutations can be solved on most calculators and computers by using the combination and permutation functions.