I am working on the problem below and just have two questions pertaining to my answers.
1) Am I clearly and correctly justfying my answers, anything I can improve on or explain better?
2) Are my answers correct?
3) For d), how can I get the pairings from a larger set of numbers? Say {1....100} and I wanted to find the pairings that add up to 103, is there an easy way to do that without counting them all?
Thanks!
Consider the rst ten positive integers: f1; 2; 3; 4; 5; 6; 7; 8; 9; 10g. We will select numbers from this
set randomly. The parity of a number is either even or odd. All answers in this problem must be
justied.
(a) How many numbers must be selected to ensure there are three odd numbers?
**8, consider the worst case where 5 even numbers are selected 3 odd numbers must then be selected**
(b) How many numbers must be selected to ensure there are three of the same parity?
**5, consider the worst case where there are 2 even and 2 odd numbers selected then one more element must be selected to have three of the same parity**
(c) If three numbers are selected, is it always the case that two of the selected numbers add up to 11?
**No, consider the worst case where 1,2,3 are select there are no two numbers that add up to 11**
(d) How many numbers must be selected to ensure there that two of the selected numbers add up to 11?
**6, consider the pairings {5,6}, {8,3}, {9,2}, {10,1}, {7,4} which all add up to 11. Since there are 5 of them, by the Pigeonhole principle, if we select 6 then we have selected two elements from the pairings**