# positive integers less than 1,000,000 have at least three digits that are the same

by Andrew
(New York)

How many positive integers less than 1,000,000 have at least three digits that are the same?

### Comments for positive integers less than 1,000,000 have at least three digits that are the same

 May 30, 2012 Positive integers less than 1,000,000 have at least three digits that are the same by: Staff Part I Question: by Andrew (New York) How many positive integers less than 1,000,000 have at least three digits that are the same? Answer: Note: 0 cannot be a first digit. 2 digit numbers (1-99) 0 = For numbers less than 3 digits (1-99). 3 digit numbers (100 to 999) A 3 digit number where: 3 digits are the same If “c” stands for the digits which are the same the possibilities are: 1 possibility: {c,c,c} --------------------- = the digit placeholder listed can only be c. This digit can be assigned only 9 possible values (0 cannot be used): 1,2,3,4,5,6,7,8,9 {1,1,1} {2,2,2} {3,3,3} {4,4,4} {5,5,5} {6,6,6} {7,7,7} {8,8,8} {9,9,9} >>Calculation: = 9 possibilities for a three digit number where at least three digits are the same --------------------- 9 = 3 digit numbers (100 to 999) numbers with exactly 3 digits (100-999). 0 cannot be included since 000 = 0 [“If” 0 could be used as the first digit, there would be 10 possibilities rather than 9 possibilities] The 9 possibilities are: {1,1,1} {2,2,2} {3,3,3} {4,4,4} {5,5,5} {6,6,6} {7,7,7} {8,8,8} {9,9,9} 4 digit numbers (1000 to 9999) A 4 digit number where: 3, or 4 digits are the same If “c” stands for the digits which are the same, and “a” stands for digits which can be different from c, the possibilities are 4 possibilities: {a,c,c,c} {c,a,c,c} {c,c,a,c} {c,c,c,a} --------------------- = the first digit placeholder listed can be either an “a”, or a “c”. However the first digit can be assigned only 9 possible values (0 cannot be used): 1,2,3,4,5,6,7,8,9 = the second digit category listed will be the remaining unused category. For example, if “a” is listed first, then “c” is listed second. The second digit category listed can be assigned 10 possible digits: 0,1,2,3,4,5,6,7,8,9 How many numbers have 3 c’s in the number (at least three digits which are the same) ? >>Calculation: = 9*10 = 90 possibilities for a four digit number where at least three digits are the same --------------------- 5 digit numbers (10000 to 99999) A 5 digit number where: 3, 4, or 5 digits are the same If “c” stands for the digits which are the same, and “a” and “b” stand for digits which can be different from c, the possibilities are ---------------------

 May 30, 2012 Positive integers less than 1,000,000 have at least three digits that are the same by: Staff --------------------- Part II 20 possibilities: {a,b,c,c,c} {a,c,b,c,c} {a,c,c,b,c} {a,c,c,c,b} {b,a,c,c,c} {b,c,a,c,c} {b,c,c,a,c} {b,c,c,c,a} {c,a,b,c,c} {c,a,c,b,c} {c,a,c,c,b} {c,b,a,c,c} {c,b,c,a,c} {c,b,c,c,a} {c,c,a,b,c} {c,c,a,c,b} {c,c,b,a,c} {c,c,b,c,a} {c,c,c,a,b} {c,c,c,b,a} --------------------- = the first digit placeholder listed can be either an a, b, or c. However the first digit can be assigned only 9 possible values (0 cannot be used): 1,2,3,4,5,6,7,8,9 = the second digit category listed will be one of the two remaining unused categories. For example, if “a” is listed first, the second digit category listed will be either b, or c. The second digit category listed can be assigned 10 possible digits: 0,1,2,3,4,5,6,7,8,9 = the third digit category listed will be the remaining unused category. For example, if “a” is listed first, “c” is listed second, “b”. The third digit category listed can be assigned 10 possible digits: 0,1,2,3,4,5,6,7,8,9 How many numbers have 3 c’s in the number (at least three digits which are the same) ? >>Calculation: = 9*10*10 = 900 possibilities for a five digit number where at least three digits are the same --------------------- 6 digit numbers (100000 to 999999) A 6 digit number where: 3, 4, 5, or 6 digits are the same If “c” stands for the digits which are the same, and “a” and “b” stand for digits which can be different from c, the possibilities are 120 possibilities: ---------------------

 May 30, 2012 Positive integers less than 1,000,000 have at least three digits that are the same by: Staff --------------------- Part III {a,b,d,c,c,c} {a,b,c,d,c,c} {a,b,c,c,d,c} {a,b,c,c,c,d} {a,d,b,c,c,c} {a,d,c,b,c,c} {a,d,c,c,b,c} {a,d,c,c,c,b} {a,c,b,d,c,c} {a,c,b,c,d,c} {a,c,b,c,c,d} {a,c,d,b,c,c} {a,c,d,c,b,c} {a,c,d,c,c,b} {a,c,c,b,d,c} {a,c,c,b,c,d} {a,c,c,d,b,c} {a,c,c,d,c,b} {a,c,c,c,b,d} {a,c,c,c,d,b} {b,a,d,c,c,c} {b,a,c,d,c,c} {b,a,c,c,d,c} {b,a,c,c,c,d} {b,d,a,c,c,c} {b,d,c,a,c,c} {b,d,c,c,a,c} {b,d,c,c,c,a} {b,c,a,d,c,c} {b,c,a,c,d,c} {b,c,a,c,c,d} {b,c,d,a,c,c} {b,c,d,c,a,c} {b,c,d,c,c,a} {b,c,c,a,d,c} {b,c,c,a,c,d} {b,c,c,d,a,c} {b,c,c,d,c,a} {b,c,c,c,a,d} {b,c,c,c,d,a} {d,a,b,c,c,c} {d,a,c,b,c,c} {d,a,c,c,b,c} {d,a,c,c,c,b} {d,b,a,c,c,c} {d,b,c,a,c,c} {d,b,c,c,a,c} {d,b,c,c,c,a} {d,c,a,b,c,c} {d,c,a,c,b,c} {d,c,a,c,c,b} {d,c,b,a,c,c} {d,c,b,c,a,c} {d,c,b,c,c,a} {d,c,c,a,b,c} {d,c,c,a,c,b} {d,c,c,b,a,c} {d,c,c,b,c,a} {d,c,c,c,a,b} {d,c,c,c,b,a} {c,a,b,d,c,c} {c,a,b,c,d,c} {c,a,b,c,c,d} {c,a,d,b,c,c} {c,a,d,c,b,c} {c,a,d,c,c,b} {c,a,c,b,d,c} {c,a,c,b,c,d} {c,a,c,d,b,c} {c,a,c,d,c,b} {c,a,c,c,b,d} {c,a,c,c,d,b} {c,b,a,d,c,c} {c,b,a,c,d,c} {c,b,a,c,c,d} {c,b,d,a,c,c} {c,b,d,c,a,c} {c,b,d,c,c,a} {c,b,c,a,d,c} {c,b,c,a,c,d} {c,b,c,d,a,c} {c,b,c,d,c,a} {c,b,c,c,a,d} {c,b,c,c,d,a} {c,d,a,b,c,c} {c,d,a,c,b,c} {c,d,a,c,c,b} {c,d,b,a,c,c} {c,d,b,c,a,c} {c,d,b,c,c,a} {c,d,c,a,b,c} {c,d,c,a,c,b} {c,d,c,b,a,c} {c,d,c,b,c,a} {c,d,c,c,a,b} {c,d,c,c,b,a} {c,c,a,b,d,c} {c,c,a,b,c,d} {c,c,a,d,b,c} {c,c,a,d,c,b} {c,c,a,c,b,d} {c,c,a,c,d,b} {c,c,b,a,d,c} {c,c,b,a,c,d} {c,c,b,d,a,c} {c,c,b,d,c,a} {c,c,b,c,a,d} {c,c,b,c,d,a} {c,c,d,a,b,c} {c,c,d,a,c,b} {c,c,d,b,a,c} {c,c,d,b,c,a} {c,c,d,c,a,b} {c,c,d,c,b,a} {c,c,c,a,b,d} {c,c,c,a,d,b} {c,c,c,b,a,d} {c,c,c,b,d,a} {c,c,c,d,a,b} {c,c,c,d,b,a} ---------------------

 May 30, 2012 Positive integers less than 1,000,000 have at least three digits that are the same by: Staff --------------------- Part IV = the first digit placeholder listed can be either an a, b, c, or d. However the first digit can be assigned only 9 possible values (0 cannot be used): 1,2,3,4,5,6,7,8,9 = the second digit category listed will be one of the three remaining unused categories. For example, if “a” is listed first, the second digit category listed will be either b, c, or d. The second digit category listed can be assigned 10 possible digits: 0,1,2,3,4,5,6,7,8,9 = the third digit category listed will be one of the two remaining unused categories. For example, if “a” is listed first, and “c” is listed second, the third digit category listed will be either b, or d. The third digit category listed can be assigned 10 possible digits: 0,1,2,3,4,5,6,7,8,9 = the fourth digit category listed will be the remaining unused category. For example, if “a” is listed first, “c” is listed second, and “d” is listed third, the final category listed will be “b”. The fourth digit category listed can be assigned 10 possible digits: 0,1,2,3,4,5,6,7,8,9 How many numbers have 3 c’s in the number (at least three digits which are the same) ? >>Calculation: = 9*10*10*10 = 9000 possibilities for a six digit number where at least three digits are the same --------------------- The grand Total >>Calculation: = 9 + 90 + 900 + 9000 = 9999 >>> the final answer 9999 positive integers less than 1,000,000 have at least three digits that are the same Thanks for writing. Staff www.solving-math-problems.com

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