1) A crazed lecturer selects three "volunteers" in any given lecture from a class of 20. No student is "volunteered" twice in a lecture. Find: a) The number of distinct groups of three which are possible. b) lf the class consists of 16 males and 4 females find the number of ways that three females can be selected in a group of 'h*u c) lf the class consists of 16 males and 4 females find the number of ways that three males can be selected in a group of three. d) lf the class consists of 16 males and 4 females find the number of ways in which two males and two females can be selected in a group of three of three. d) lf the class consists of 16 males and 4 females find the number of ways in which more males than females are selected in a group of three. (7 marks) a2) a) Use Euclid's algorithm to find the greatest common divisor of 7 and 51- b) Find the multiplicative inverse of 7 mod 51. c) Hence solve the linear congruence 7x = 3 (mod 51). 3) Solve the equation 7101 working. 4) A function is defined as : xmod 51. State any theorem's used and show all {(-1,0), (0, a) State the domain and range of that function. -3), (2 ,-3), b) The co-domain is implicitly defined. ls the function onto? c) ls the function injective? Explain. (3,0), (4,5)} d) ls the function invertible? Explain. 6) a) ln choosing five letters in any order from the word SMOKING, and assuming each choice is equally likely, what is the probability of choosing (The vowels are A, E, l, O U)? b) i)ln how many ways can the numbers 1, 2, 3, 4, 5, 6,7, 8 be arranged around a circle? ii)How many of these arrangements have at least three even numbers together? just one vowel. 7) a) From an ordinary deck of cards, we remove oneeard. What'ls the probability that it is a Queen, is a diamond, or is red? b) lf the universal set t = {1,2,3,4,5,6,7,8,9,10} draw a Venn diagram to represent the evens, the multiples of three and the primes. Label your diagram carefully. I 8) a) Let A -- {(x,y)lx,y (a,b)R(c,d) * a* c i) Reflexive ii) Symmetric integers} Define a relation on A by the rule = b + d. Determine if the relation is iii) Transitive b) Explain why the relation is or is not an equivalence relationship.
