Answer
a) 990
b) 550
c) 27
Work Step by Step
Maximum possible strings with three decimal digits is 1000, since there is a choice of 10 digits for each of the three positions in the string.
a) This can be found by subtracting the total number of possible strings by all possible strings that violate the condition (do not have same digit three times) i.e, that have same digit three times.
Clearly there are 10 strings that consist of the same digit three times ( 000, 111, ... , 999).
Therefore there are 1000 - 10 = 990 strings that do not.
b) If we begin our string with an odd digit, then we have 5 choices for this digit. The rest two digits still have 10 choices each.
Therefore there are 5 · 10 · 10 = 500 such strings.
c) Here we can first choose the position of the digit that is not a 4 (3 ways) and this digit can be chosen in 9 ways (leaving 4) .
Therefore there are 3 · 9 = 27 such strings.
