1. For all parts of this question, working is required, including combinatoric/factorial notation as needed as well as final answers. An answer consisting of solely an integer will be awarded 0 marks.

a) A committee of 9 persons is to be chosen from 7 lecturers and 10 students so as to contain at least 2 lecturers and 5 students. Find the number of ways a committee can be formed

i) Containing at least 2 lecturers and 5 students

ii) If 5 particular students are always present in the committee

iii)When 5 particular students are never together

b) A traveller forgets their passcode consisting of four digits. However, he remembers that his passcode has the following digits in it: 3, 2, 7, 4. Find the total number of trials necessary to obtain the correct passcode:

i) If a number is repeated

ii) If a number is not repeated

c) Four dice are rolled. Find the number of possible outcomes in which at least one die shows a 6.

d) There are twelve units titled COS1, COS1, COS3,...COS12. Out of the twelve units, six units are to be arranged in such a way COS1 must occur whereas COS4 and COS5 do not occur. Find the number of such possible arrangements.

e) There are seven pens in drawer A and seven notebooks in drawer B. You have been asked to match each pen in drawer A with a notebook in drawer B. What is the total number of ways this can be done?