Problem

The number 43210 contains the combination "32"; the numbers 43120 and 42310 do not contain the combination "32." How many 5digit numbers (between 10000 and 100000 contain the combination "32"?

Answer

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Answer

\(\boxed{2781}\)

Steps

Step 1 :Consider the possible positions of '32' in the 5-digit number and count the number of possibilities for the remaining digits.

Step 2 :Case 1: 32XXX, there are 10 choices for each of the remaining 3 digits (0-9), so there are 10 * 10 * 10 = 1000 possibilities.

Step 3 :Case 2: X32XX, there are 9 choices for the first digit (1-9) and 10 choices for each of the remaining 2 digits (0-9), so there are 9 * 10 * 10 = 900 possibilities.

Step 4 :Case 3: XX32X, there are 9 choices for the first digit (1-9) and 10 choices for the second digit (0-9) and 10 choices for the last digit (0-9), so there are 9 * 10 * 10 = 900 possibilities.

Step 5 :Subtract the numbers that have two occurrences of '32':

Step 6 :Case 1: 3232X, there are 10 choices for the last digit (0-9), so there are 10 possibilities.

Step 7 :Case 2: X3232, there are 9 choices for the first digit (1-9), so there are 9 possibilities.

Step 8 :Calculate the total number of 5-digit numbers containing '32': total = 1000 + 900 + 900 - 10 - 9 = 2781

Step 9 :\(\boxed{2781}\)

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