6. (10 points)The mayor of city X wants to assign a unque identification number

Question

6. (10 points)The mayor of city X wants to assign a unque identification number with n symbols to each person living in city X such that each symbol can be either a digit or one of the letters A, B, C, D, E, or F. Assuming that the population of city X is 16,800,000, what is the smallest possible value for n? (Hint: First find out how many different identification numbers can we make using n symbols; then use the Pigeonhole Principle to find an inequality for n) 7. (5 points) In a set of 150 integers, 120 of them are divisible by 3, 60 of them are divisible by 5, and 40 of them are divisible by 15. How many integers are divisible by neither 3 nor 5 (Hint: An integer is divisible by 15 iff it is ivisible by both 3 and 5) 8. (15 points) Let n be a randomly chosen 7-digit positive integer (a) What is the probability that n is even? (b) What is the probability that n 7,000, 000? (c) What is the probability that 1,500, 000 n 6, 200,000? (d) What is the probability that n does not contain repeated digits and is not made of digits 0 and 5? (e) What is the probability that n has no repeated digits and is divisible by 5? 9. (15 points) In a box of 25 light bulbs, 15 of them are defective and the rest are working (a) In how many different ways can we pick 7 light bulbs from the box? (b) If we randomly pick eight light bulbs, what is the probability of picking no defec- tive light bulbs? (c) If we randomly pick nine light bulbs, what is the probability of picking five or seven defective light bulbs?

Explanation / Answer

Solving only Q1) and Q2)as per Chegg rules:

A)no of total unique identification numbers= 6^n

To find minimum n,

6^n > 16800000 -(1) and 6^(n-1) < 16800000 -(2)

Solving we get n=10

2) 120 are divisible by 3

60 are divisible by 5

40 are divisible by 15, that is 40 are divisible by both 3,5

Only divisible by 3=120-40=80

Only divisible by 5=60-40=20

Total divisible by 3or5= 80+20+40=140

Not divisible by 3 and 5= 150-140=10

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