Math IOQM

Q1.  Show that there does not exist a function (f: N -> N) which satisfies (a) f(2)=3 (b) f(mn) = f(m).f(n) for all m,n in N (c) f(m) < f(n) whenever m < n. S1.

Q1. Objects A and B move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object A starts at (0,0) and each of its steps is either right or up, both equally likely. Object B starts at (5,7) and each of its steps is either left or down, both equally likely. What is the probability that the objects meet? S1. Total distance between them: 12 Each step reduces distance by 2. So steps needed to meet = 6 During 6 steps, x co ordinate of A can go all the way upto 6. But B can't reach 6 since it's going left from 5. So they can meet at only x = 0,1,2,3,4,5 What is the probability that A's x co-ordinate is 'k' after 6 steps? 6Ck * p^k * (1-p)^(6-k) where p is probability that A will move right and (1-p) is the probability of moving up. Here p = 1/2 So it becomes 6Ck * (1/2)^6. How does B reach x = k? It has to move left by 5-k. k = 5 - (5 - k) What is the probability that out of 6 steps taken by B, 5-k are to the left? 6C(5-k) * (1/2)^6 Sinc...

Q1. In a group of 20 students: 5 are Yellow House 7 are Red House 4 are Blue House 4 are Green House How many ways can you choose: (i) 12 kids, such that exactly 3 kids from each house are present. (ii) 10 kids, such that at least 2 kids from each house are present. (iii) 15 kids, such that at least 3 kids from each house are present. S1. Attempt 1: (i) 5C3 * 7C3 * 4C3 * 4C3 (ii) 5C2 * 7C2 * 4C2 * 4C2 * 12C2 (iii) 5C3 * 7C3 * 4C3 * 4C3 * 8C3 In the above attempt, (i) is correct. But (ii) and (iii) are overcounting. Why? In (ii), Let's say we pick A,B from yellow house and the during 12C2 pick C from yellow house.  We also have counted the case where we first pick A,C from yellow and then pick B during 12C2. So this is wrong. Same for (iii). Another thought: to use stars and bars where houses are buckets and students are objects. But for stars and bars, the objects have to be same to each other. Here they are distinct. But stars and bars can tell us how many ways are there to group ...