Consider an 21-ball lottery game. In total there are 21 balls numbered 1 through to 21 inclusive. 6 balls are drawn (chosen randomly), one at a time, without replacement (so that a ball cannot be chosen more than once). To win the grand prize, a lottery player must have the same numbers selected as those that are drawn. Order of the numbers is not important so that if a lottery player has chosen the combination 16, 17, 18, 19, 20, 21 and, in order, the numbers 18, 16, 21, 20, 17, 19 are drawn, then the lottery player will win the grand prize (to be shared with other grand prize winners). You can assume that each ball has exactly the same chance of being drawn as each of the others.

(a) Consider a population of size N = 21. How many different random samples of size n = 6 are possible from a population of N = 21? (2 marks)

(b) Suppose that you choose the numbers 16, 17, 18, 19, 20 and 21 ahead of the next lottery draw. As a fraction, what is the exact probability that you will win the grand prize in the lottery in the next draw with these numbers? (2 marks)

(c) Continuing on part (b) and again as a fraction, what is the exact probability that you will not win the lottery in the next draw with these numbers? Show workings. (2 marks)

(d) This question is tougher and you need to think carefully about the answer. Recall that the order of the numbers chosen is not important and that each number can only be chosen once. In total, how many combinations are there available that include the numbers 20 and 21 but not the numbers 16, 17, 18 and 19? Explain and show workings. (Hint: The available combinations must include the numbers 20 and 21 and they may also include the numbers 16, 17, 18 or 19, but they must not include the numbers 16, 17, 18 and 19 in one combination at the same time.)