Six three-digit numbers contain only the digits 5 and 6. This is assuming you mean both the digits of 5 and 6. If not than 8, as it would include 555 and 666. 556, 565, 566, 655, 656, 665
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Since the numbers begin with 123, they all contain a 3. As a result, they all "contain 3 or 7". So the remaining 4 digits can be any digits and this gives 104 = 10,000 such numbers.
874
Most credit cards in the United States contain 16 digits broken into four groups of four digits. American Express cards contain 15 digits, however.
10 digits are numbers in the billions.
Ten different digits can be used to make 10C4 = 10*9*8*7/(4*3*2*1) = 210 four-digit numbers. Either numbers starting with 0 are permitted or the 10 digits do not contain a 0.