Solve 6-Digit Combinations Question with 0-9

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Discussion Overview

The discussion revolves around calculating the number of possible 6-digit codes Jacob can create for his bank account using the digits 0 to 9, with specific conditions that the first digit must be odd and the last digit must be even, and that no digits can be repeated.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • Jacob has 5 choices for the first digit (odd numbers: 1, 3, 5, 7, 9) and 5 choices for the last digit (even numbers: 0, 2, 4, 6, 8).
  • Some participants suggest calculating the total combinations by filling in the first and last digits first, followed by the remaining digits, leading to a proposed formula of (5)(9)(8)(7)(6)(4) or variations thereof.
  • There is a discussion about the number of choices available for the second, third, and fourth digits after selecting the first and last digits, with one participant noting that this would leave 8 choices for the second digit, 7 for the third, and so on.

Areas of Agreement / Disagreement

Participants generally agree on the approach of selecting the first and last digits first, but there are variations in the proposed calculations and the exact number of choices for the remaining digits, indicating some uncertainty in the final answer.

Contextual Notes

There are unresolved aspects regarding the exact calculation method and the implications of the choices made for the remaining digits after the first and last have been selected.

greprep
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"Jacob needs to create a 6-digit code for his bank account using the digits from 0 to 9. He wants the first digit to be odd and the last digit to be even. If he does not repeat any digits, how many different 6-digit codes could Jacob create?"

Would the best way to solve this be: (5)(9)(8)(7)(6)(4)?
 
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greprep said:
"Jacob needs to create a 6-digit code for his bank account using the digits from 0 to 9. He wants the first digit to be odd and the last digit to be even. If he does not repeat any digits, how many different 6-digit codes could Jacob create?"

Would the best way to solve this be: (5)(9)(8)(7)(6)(4)?

You're close, but I would begin by filling in the first and last digits first (5 choices for each), and then fill in the remaining digits:

$$N=(5)(8)(7)(6)(5)(5)=\,?$$
 
Oh, I see. So I would put in 5 first for the potential odd number (1,3,5,7,9), then 5 at the end for the potential even numbers (0,2,4,6,8), and that leaves us with 10-2 (8) choices for the second, 7 for the third, 6 for the 4th, etc?
 
greprep said:
Oh, I see. So I would put in 5 first for the potential odd number (1,3,5,7,9), then 5 at the end for the potential even numbers (0,2,4,6,8), and that leaves us with 10-2 (8) choices for the second, 7 for the third, 6 for the 4th, etc?

Yes, that's what I was suggesting. :)
 

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