Billy chooses three dfferent non-zero digits and makes six different 3-digit numbers with no digit repeated in any number. He notices that one of the 3-digit nubers N say is the aerage of the other five. Find all possible values of N.
Three different non-zero digits: a, b, c
Hint: How many different 3-digit numbers can you make with only a, b, and c?
pls hepl me
6
that is 3 factorial
a b c
a c b
b a c
b c a
c a b
c b a
To solve this problem, let's think step by step.
First, we need to consider the possible combinations of three different non-zero digits. Since there are 9 digits to choose from (1-9), there are a total of ${9 \choose 3} = 84$ different combinations possible.
Next, we need to make the six different 3-digit numbers from these three digits, without repeating any digits. To do this, we can think of arranging the three digits in different orders.
Let's assume the three digits chosen are A, B, and C, where A < B < C.
The possible arrangements would be:
ABC, ACB, BAC, BCA, CAB, CBA
Now, we know that one of these six numbers, say N, is the average of the other five numbers. This means that N should be the sum of the other five numbers divided by 5.
To find all possible values of N, we need to try all 84 combinations from step 1 and check for each combination if any of the six arrangements satisfies the condition.
Let's analyze how to check if a given arrangement satisfies the condition of N being the average of the other five numbers:
We know that the sum of the six numbers is even (because it is the sum of five odd numbers and N). So, N must be even too.
Now, for each arrangement, we can find the sum of the other five numbers and divide it by 5 to check if it equals N. If it does, we add N to our list of possible values.
By going through all the combinations and checking the arrangements, we can find all the possible values of N.
Note: Since we are dealing with a relatively small number of combinations, it is feasible to manually calculate the possible values of N. However, it would require a bit of computation and checking. To simplify things further, we can use a programming language or a spreadsheet program to automate the process and get the results quickly.