Question

I’m thinking of a 4-digit number. If I add its four digits together I get 34. How many different numbers are there that I could possibly be thinking of?

Answer 1

To solve this problem, we need to find the different 4-digit numbers whose digits add up to 34. We can use the concept of permutations to calculate the number of combinations for each possible thousands digit. Adding up the number of combinations for each thousands digit, we get 43 different numbers.

Step-by-step explanation:

• To solve this problem, we need to find the different 4-digit numbers whose digits add up to 34. Let's start by considering the largest possible digit, which is 9. If we use 9 as the thousands digit, then the sum of the remaining three digits needs to be 34 - 9 = 25.
• We can distribute these 25 units among the three digits (hundreds, tens, and ones) in different ways.
• To calculate the number of combinations, we can use the concept of permutations. There are 25 units to be distributed among 3 slots, which can be done in 3! (3 factorial) ways.
• So, there are a total of 3! = 3*2*1 = 6 different numbers where the digits add up to 34 when the thousands digit is 9.
• Following the same logic, if we use 8 as the thousands digit, then the sum of the remaining three digits needs to be 34 - 8 = 26.
• Again, there are 26 units to be distributed among 3 slots, which can be done in 3! = 6 ways.
• We can continue this process for each possible thousands digit, 7, 6, 5, 4, 3, 2, and 1.
• Finally, if we use 0 as the thousands digit, then the sum of the remaining three digits needs to be 34 - 0 = 34.
• There is only one way to distribute 34 units among 3 slots (34, 0, and 0).
• Adding up the number of different combinations for each thousands digit, we get 6 + 6 + 6 + 6 + 6 + 6 + 6 + 1 = 43.
• Therefore, there are a total of 43 different numbers that could possibly be the student's 4-digit number.