Question:

Step 1: Identify the digits and their frequencies in the number 222335588. The digits are: 2 (appears 3 times) 3 (appears 2 times) 5 (appears 2 times) 8 (appears 2 times) There are a total of 9 digits. Step 2: Categorize the digits as odd or even. Odd digits: 3, 3, 5, 5 (4 odd digits in total). Even digits: 2, 2, 2, 8, 8 (5 even digits in total). Step 3: Identify the positions in a nine-digit number. A nine-digit number has positions 1, 2, 3, 4, 5, 6, 7, 8, 9. Even positions are: 2, 4, 6, 8 (4 positions). Odd positions are: 1, 3, 5, 7, 9 (5 positions). Step 4: Determine the number of ways to place the odd digits in the even positions. There are 4 odd digits (3, 3, 5, 5) to be placed in the 4 available even positions. The number of distinct arrangements for these repeated digits is given by the multinomial coefficient formula: $$ \frac{4!}{2! \cdot 2!} = \frac{24}{4} = 6 $$ Step 5: Determine the number of ways to place the even digits in the odd positions. There are 5 even digits (2, 2, 2, 8, 8) to be placed in the 5 available odd positions. The number of distinct arrangements for these repeated digits is given by the multinomial coefficient formula: $$ \frac{5!}{3! \cdot 2!} = \frac{120}{6 \cdot 2} = \frac{120}{12} = 10 $$ Step 6: Calculate the total number of different nine-digit numbers. The total number of different nine-digit numbers is the product of the number of ways to arrange the odd digits in even positions and the number of ways to arrange the even digits in odd positions: $$ 6 \times 10 = 60 $$