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?
1957, 1975
Can you find any others?
yes. there are a total of 10.
thanks for the help
I'm curious. What are the other years that fulfill these requirements?
To find the number of years in the 20th century that have four different odd digits and a sum of digits greater than 20, we can follow these steps:
Step 1: Determine the range of years in the 20th century.
The 20th century spans from the year 1900 to 1999, inclusive.
Step 2: Identify the years with four different odd digits.
To have four different odd digits, the year must consist of four odd numbers that are not repeated. The possible digits that are odd are 1, 3, 5, 7, and 9.
Step 3: Calculate the number of ways to select four different odd digits.
Using the concept of combinations, we can calculate the number of ways to select four different odd digits from the set {1, 3, 5, 7, 9}. This can be represented as C(5, 4) and is equal to 5.
Step 4: Determine the sum of the digits for each year.
For each arrangement of four odd digits, we need to calculate the sum of those digits. To do this, start by fixing the thousands digit to be one of the four odd digits available. Then, select the remaining three digits from the remaining three odd digits in C(4, 3) different ways.
However, we have the additional constraint that the sum of the digits must be greater than 20.
Step 5: Calculate the number of valid arrangements.
To find the number of valid arrangements, we need to count the possible ways to form a number greater than 20 using the remaining three digits (excluding the thousands digit). Since the remaining three digits are all odd, the smallest number we can create is 3 + 5 + 7 = 15. Therefore, the valid sums will be in the range of 21 to 27.
Step 6: Count the number of years that satisfy the condition.
For each arrangement of the three remaining digits, we can calculate the number of valid sums within the range of 21 to 27, inclusive. By summing up these counts for each arrangement, we will obtain the total number of years that satisfy the condition.
After performing these steps, you will have the answer to the question.