MAT 344Challenge Set 1The following questions illustrate some interesting mathematical counting ideas whichwill be the topic of future material in this course. Each question can be answered bysimply writing down all of the possible cases step by step. Done correctly, this will leadyou to the general principle involved. Try to prove your answer carefully.1. Determine the number of positive integers less than or equal to 100 which are:- odd (that is, not divisible by 2)- not divisible by 2 or 5- not divisible by 2, 5, or 112. Find the number of ways of giving 4 different colour pens to 2 people a) without anyrestrictions b) so that each person gets at least one pen.3. How many different addresses on Memory Lane can be constructed using the digits 0through 9, assuming that no address can have more than 2 digits? ( Note: Think carefullyabout what constitutes an address.)4. Show that in any group of 13 people, at least two were born in the same month. ( O.K.,
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