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A number lock contains 6 digits. How many different zip codes can be made with the digits 0–9 if repetition of the digits is allowed upto 3 digits from the beginning and the first digit is not 0?

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A number lock contains 6 digits. How many different zip codes can be made with the digits 0–9 if repetition of the digits is allowed upto 3 digits from the beginning and the first digit is not 0?

(a) 254307

(b) 453600

(c) 458760

(d) 972340

The question was posed to me during an online exam.

My question is taken from Counting topic in chapter Counting of Discrete Mathematics

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The correct option is (b) 453600

To explain I would say: For the first position, there are 9 possible choices (since 0 is not allowed). After that number is chosen, there are 10 possible choices (since 0 is now allowed) for the second digit, for the third digit there are 10 possible choices, 9 possible choices for the fourth digit and 8 possible choices for the fifth digit and 7 possible choices for the sixth digit. The count of number locks = 453600.

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