Question

Show steps, answer is 13440

3. How many odd five-digit numbers have all the digits different? Explain how you arrived at your answer.

Answer 1

Let's understand the odd five-digit number first. It means that the last number must be odd (1,3,5,7,9) whereas the first four can be any. So let’s think about the possible options (0,1,2,3,4,5,6,7,8,9) and how we can fill the spots: 

Let the spots be #1, #2, #3, #4, #5

_ _ _ _ _

As I mentioned, last spot # (spot #5) must be an odd number. So let’s say one of these odd numbers (1,3,5,7,9) was put at the end.

Spot #1: Can’t be zero because then the number won’t be five-digit, right? Ex: 05647 is not a five-digit number, it becomes four-digit. And since one of the odd numbers is already at spot#5 (last spot), that leaves us with only 8 choices for spot#1. Agree? 

Spot #2: Now spot 2 can have 0 because 0 in the middle won’t affect the number (meaning it wouldn’t change the number from being a 5 digit number to 4 digit number like it did in the first spot). But again, one of the numbers is already in the last spot and the first spot is also occupied now, therefore, we again have 8 choices for our spot #2 (2 spots filled with 2 numbers out of 10).

Spot #3: 7 choices

Spot #4: 6 choices

Answer: 8 x 8 x 7 x 6 =13440.

Answer 2

Please don't hesitate to give a "thumbs up" for the answer in case this has helped you.

odd 5 digit numbers is :

_ _ _ _ _

The last blank has to be 1 of the odd numbers, which can happen in 5 ways as there are 5 odd numbers {1,3,5,7,9}

So,

_ _ _ _ 5

Now, Out of the total 9 numbers left (1 odd number is already taken up in the last blank), we can have 8*8*7*6 combinations.

(8 in the first blank because we can't have 0 here, then 8 are remaining - so in the 2nd blank 8 type of numbers can come, then 7 , then 6)

So, _ _ _ _ 5 becomes 8*8*7*6*5

= 13440 ways of having odd 5-digit numbers have all the digits different.

Answer 3

EDITED:

Let's understand the odd five-digit number first. It means that the last number must be odd (1,3,5,7,9) whereas the first four can be any. So let’s think about the possible options (0,1,2,3,4,5,6,7,8,9) and how we can fill the spots: 

Let the spots be #1, #2, #3, #4, #5

_ _ _ _ _

As I mentioned, last spot # (spot #5) must be an odd number. So let’s say one of these odd numbers (1,3,5,7,9) was put at the end.

Spot #1: Can’t be zero because then the number won’t be five-digit, right? Ex: 05647 is not a five-digit number, it becomes four-digit. And since one of the odd numbers is already at spot#5 (last spot), that leaves us with only 8 choices for spot#1. Agree? 

Spot #2: Now spot 2 can have 0 because 0 in the middle won’t affect the number (meaning it wouldn’t change the number from being a 5 digit number to 4 digit number like it did in the first spot). But again, one of the numbers is already in the last spot and the first spot is also occupied now, therefore, we again have 8 choices for our spot #2 (2 spots filled with 2 numbers out of 10).

Spot #3: 7 choices

Spot #4: 6 choices

Spot #5: 5 choices

Answer: 8 x 8 x 7 x 6 x 5=13440