9. a) i) There are 2 awards being given out and ten students to give it to. So there are 10 possibilities to give the first award, and though the same person can get both awards there are another 10 possibilities to give out the second award, so the total number of possibilities is 10x10=100
a) i) This is almost the same, 10 possibilities for the first award to be given, but though that same person cannot receive the award again, there are only 9 possibilities for the second award to be given out. So there are 10x9=90 possibilities for the second award.
b) i) Lets find out how many possibilities there are for this particular student to receive a distinction. So let's give him the distinction, so there is 1 arrangement of who can get the distinction, and though he can't get the merit, there are 9 possibilities for who gets the merit. All together there are 1x9=9 arrangements for the two awards if this student gets the distinction. So to find the probability that he gets the distinction, we take the number of possibilities that he does get the distinction and divide it by the total number of arrangements, so the probability is 9/90=1/10, which is 10%.
b) ii) Similarly, we give the student the merit, so there is one arrangement for who gets the merit, and then there are 9 possibilities for who gets the distinction. So again, in total there are 1x9=9 possibilities for the student receiving the merit, so to find the possibility we take the possibilities that he gets the merit and divide it by the total number of possibilities, and we get 9/10=1/10 again.
b) iii) We take the probably that he receives a merit or a distinction and subtract that away from 1. So we get 1 - 1/10 - 1/10= 8/10=4/5. Another way to do this is, we don't want the student to receive any awards, so we give we take him out of the number of people who can receive awards, so there are now 9 possible award recipients. To find the total number of arrangements for this group receiving the award, we do the same thing as we did in question 1. There are 9 people to give the first award, and as they cannot receive both, 8 people to give the second award, so there are 9x8=72 possibilities for giving out the award with this particular student not receiving any awards. To find the possibility he does not receive any awards, we take the number of possibilities when he doesn't receive an award and divide it by the total number of possibilities, so 72/90=4/5.
c) We already have the total number of arrangements for if the student can receive both awards, which is 100. So for the particular student receiving at least 1 award, there are 2 cases. He gets 1 award or he gets 2. For getting 1 award, we give him the award, so 1 student can receive it, and there are another 9 students to give the next award to (not 10, as then he could receive two awards). So the number of arrangements is 1x9=9. To find the number of arrangements that he receives both is fairly simple, we give him both awards so there is 1x1=1 arrangements. So to find the possibility that he receives at least one award, we add these cases' possibilities up, which is ten, and divide it by the total number of possibilities, so we get 10/100=1/10.
Hope I helped.
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