(a) How many positive integers are there between 1000 and 4999, inclusive?(b) How many positive integers between 1000 and 4999, inclusive:1. have no repeated digit?2. have at least one repeated digit?3. have at most two repeated digits?Note that by `one repeated digit' we mean that there is a digit that appearsat least twice (eg, 1123 has one repeated digit). Similarly, by `two repeated digits'we mean a digit that appears at least three times (eg, 1112).

Answer:a) 4000b1) 480b2) 3,520b3) 1,434Step-by-step explanation:a) 4999 - 1000 +1 = 4000 b) We must find numbers of the form “abcd” where “a” is 1,2,3 or 4 and all the digits are different to each other. The digits “bcd” must be chosen from the set {0,5,6,7,8,9} with no repetition. So by the fundamental rule of counting, we have 4*6*5*4 = 480 numbers between 1000 and 4999 with no repeated digits. 2. We can see that this set is the complement of the previous one, so there are 4,000 - 480 = 3,520 3. Let's compute the amount of number with exactly 2 repeated digits and add it to 480 (amount with no repeated digits) Starting with 1 we have numbers of the form 11xy, 1x1y and 1xy1 where xy are permutations of the set {0,2,3,4,5,6,7,8,9} taken 2 at a time, so we have 72+72+72 = 216. The same procedure is repeated for 2,3 and 4 and we have  216+216+216 +216 = 864 numbers starting with 1,2,3 or 4 and exactly two repeated digits. Now we found the triplets xxy, xyx, yxx with exactly 2 repeated digits none of them 1,2,3 or 4 and we have  15+15+15+15+15+15 = 90 And finally we have 864+90+480 = 1,434 digits with at most 2 repeated digits.