Hello!
Here it is important to begin with a clear understanding of the question
By distinct digits, it is meant that the digits in the number do not repeat
For instance, 222,231 cannot be one of the possibilities because the number 2 repeats too often. Each of the digits in the ones, tens, hundreds, etc. position must be distinct from all the other digits.
We are given 6 different numbers 1, 2, 3, 4, 5, and 6 to construct with
We are told that the number must be positive, an integer, and odd
This means that the number can only end in 1, 3, or 5
Let's go methodically through the possibilities
How many 1 digit numbers can we make?
1, 3, and 5 = 3 different numbers that are positive odd integers
How many 2 digit numbers can we make?
We know that the ones value will be one of three numbers, either 1, 3, or 5
That means that for the tens position, 5 numbers will be left over to choose from and combine with the other number:
5 x 3 = 15 different possible combinations or distinct numbers
How many 3 digit numbers can we make?
After using a number in the ones positon and another number in the tens position, we are left with four numbers to choose from for the hundreds position
4 x 5 x 3 = 60 different possible combinations
How many 4 digit numbers can we make
There are 3 numbers to choose from left over for the thousands position
3 x 4 x 5 x 3 = 180 different possibilities
How many 5 digit numbers?
There are 2 numbers left to choose from to make sure that each digit is distinct from the others so we get:
2 x 3 x 4 x 5 x 3 = 360 different possibilities
And finally, how many 6 digit numbers can we make?
Well, there will only be 1 number left over to put in the hundred thousands position:
1 x 2 x 3 x 4 x 5 x 3 = 360 possibilities
Now we add up all the different numbers that can be made:
360 six digit numbers + 360 five digit numbers + 180 four digit numbers + 60 three digit numbers + 15 two digit numbers + 3 one digit numbers =
978 different numbers can be made
The answer is (D)
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