Some years in the 20th century have 4 different odd digits. How many of those years have the sum of the digits greater than 20?
To solve this problem, we need to find the number of years in the 20th century that have 4 different odd digits (0, 2, 4, 6, 8 are not considered odd digits in this case) and a sum of digits greater than 20.
First, let's determine the range of years in the 20th century. The 20th century encompasses the years from 1900 to 1999.
Now, let's break down the problem step by step:
Step 1: Identify the possible odd digits that can be used in the year number. There are 5 odd digits: 1, 3, 5, 7, 9.
Step 2: Determine the number of ways we can choose 4 different odd digits from the 5 available options. We can calculate this using the combination formula, which is given by nCr = n! / (r! * (n - r)!), where n is the total number of options and r is the number of choices. In this case, we have 5 options and we want to choose 4, so the calculation is 5! / (4! * (5 - 4)!), which simplifies to 5.
Step 3: Calculate the total number of years in the 20th century that have 4 different odd digits. Since we have 5 options for the first digit, 4 options for the second digit, 3 options for the third digit, and 2 options for the last digit (since no digit can be repeated), we can multiply these options together: 5 * 4 * 3 * 2 = 120.
Step 4: Determine the number of years from Step 3 where the sum of the digits is greater than 20. To do this, we need to analyze the different possibilities.
- If the first digit is 9, the remaining three digits can be any of the other 4 options, resulting in 4 * 4 * 3 = 48 possibilities.
- If the first digit is 7, the second digit must be 9 to ensure a sum greater than 20. The remaining two digits can be any two from the other 3 options, yielding 3 * 2 = 6 possibilities.
- If the first digit is 5, the second digit can be either 9 or 7, resulting in 2 options. The remaining two digits can be any two from the remaining 3 options, giving us 3 * 2 = 6 possibilities.
- If the first digit is 3, it must be paired with both 9 and 7 to achieve a sum greater than 20. The last remaining digit can be any one of the 2 remaining options, leading to 2 possibilities.
- If the first digit is 1, it must be combined with 9, 7, and 5 to ensure a sum greater than 20. The last digit can be any of the remaining 4 options, resulting in 4 possibilities.
Adding up all the possibilities: 48 + 6 + 6 + 2 + 4 = 66.
Therefore, there are 66 years in the 20th century that have 4 different odd digits and a sum of digits greater than 20.