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Odometer Full House

In poker, a full house is three of a kind of one card and a pair of another card (for example, 3 aces and 2 kings).

If an automobile odometer shows five digits and the leading digit is not 0, what is the probability the digits comprise a full house (for example, three 7s and two 8s in any order)?

Be the first to turn in the correct answer to Mr. Day and win a candy bar.

Solution:

1-digit numbers: all nine are palindromes.

2-digit numbers: 11,22,33,...,99 are the nine palindromes

3-digit numbers: 9 choices for the first digit (which will also be the third digit) and 10 choices for the middle digit. Thus, a total of 90 palindromes.

4-digit numbers: 9 choices for the first digit and 10 choices for the second digit, which also determine the 4th and 3rd digits, respectively. So, again a total of 90 palindromes.

5-digit numbers: 9 choices for the first digit, 10 choices for the second digit, and 10 choices for the middle digit (5th and 4th digits are determined by the 1st and 2nd digits, respectively). This gives a total of 900 palindromes.

6-digit numbers: 9 choices for the first digit, 10 choices for the second digit, and 10 choices for the third digit (6th, 5th and 4th digits are determined by the 1st, 2nd and 3rd digits, respectively). This gives a total of 900 palindromes.

Therefore, the total number of odometer palindromes is 2(9+90+900)  = 1998.

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