A raffle ticket has an ID that is a sequence of 12 digits. We wish to determine

Question

A raffle ticket has an ID that is a sequence of 12 digits. We wish to determine how many such IDs contain the each of odd digits at least once.

(a). Explain why the following “solution” is wrong: First place the 1, 12 ways, then place the 3, 11 ways, then place the 5, 10 ways, place the 7, 9 ways, place the 9, 8 ways, finally pick any of the 10 digits to go in any of the remaining 7 spots (order important, repeats allowed) 107 , giving 12 · 11 · 10 · 9 · 8 · 107 .

(b). Solve the problem correctly!

Explanation / Answer

a) It is wrong because zero cannot go in the first spot.

b) We subtract the cases in which a zero is picked in the first slot -

12 * 11 * 10 * 9 * 8 * 10^7 - 12 * 11 * 10 * 9 * 8 * 1 * 10^6 = 12 * 11 * 10 * 9 * 8 *9*10^6 = 855360000000

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