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# Counting

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How many 4-digit numbers are there where any two consecutive digits have different parity?

May 7, 2022

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Here is my attempt:
So suppose the 4 digits are:  _ _ _ _
The first _ could be even or odd.

If it is even then,

let Case (1): EVEN ODD EVEN ODD​

If it is odd then,

let Case (2): ODD EVEN ODD EVEN
Even numbers: 0,2,4,6,8

Odd numbers: 1,3,5,7,9

For Case (1):

Since the first digit _ could be 2,4,6,8 then there are 4 possibilities.

The second digit is odd, so it could be 1,3,5,7,9 then there are 5 possibilities.

The third digit is even, so it could be 0,2,4,6,8 (Notice, the first digit couldn't be 0, as that would make the number "3-digit") then there are 5 possibilities.

The fourth digit is odd, so it could be 1,3,5,7,9 then there are 5 possibilities.

Thus multiplying all the possibilities:

4*5*5*5 = 500       (*)

For Case (2):

Same method as Case (1), except the first digit is odd so it could be: 1,3,5,7,9 then there are 5 possibilities
5*5*5*5 = 625  (**)