Find the cardinality of the set of all five digit numbers th

Find the cardinality of the set of all five digit numbers that have a digit that is repeated three times in a row. Do not allow for leading zeros, so, for example, neither 23 nor 2229 are in the set. Explain or show your work.

Solution

we have to make set of 5 digit numbers whose first digit can\'t be 0 and same digit repeates consecutively thrice

there can be 3 different cases:

case 1:

say repeating digit is on first place from left then due to consecutivity first three digits from left will be same

and can\'t be 0

so there are 9 ways to fill first 3 digits

remaining 4th and 5th place can be any number from 0 to 9 means each of 4th and 5th position can be filled in 10 ways

so total number of this configuration =9*10*10=900

--------------------------------

case 2:

say repeating digits is on 2nd,3rd and 4th place

so these 3 places can be filled in 10 ways as they can be any of 0 to 9

first digit can\'t be 0 there are 9 ways to fill first digit

5th digit can be any number so that can be filled in 10 ways

so total number of this configuration =9*10*10=900

---------------------------

case 3:

say repeating digits is on 3rd, 4th and 5th place

so these 3 places can be filled in 10 ways as they can be any of 0 to 9

first digit can\'t be 0 there are 9 ways to fill first digit

2nd digit can be any number so that can be filled in 10 ways

so total number of this configuration =9*10*10=900

-------------

adding up all those cases gives 900+900+900=2700

Hence given set will contain 2700 numbers

so cardinality of that set will be 2700