Adjacent Digits

How many six-digit numbers do not have adjacent digits the same? That is, we are excluding numbers like 689113.
Source: NCTM Mathematics Teacher

SOLUTION
Let abcdef represent the six-digit numbers.
Choices for a=\left \{any\;digit\;except\;0\right \}\rightarrow 9 choices
Choices for b=\left \{any\;digit\;except\; a\right \}\rightarrow 9 choices
Choices for c=\left \{any\;digit\;except\; b\right \}\rightarrow 9 choices
Choices for d=\left \{any\;digit\;except\; c\right \}\rightarrow 9 choices
Choices for e=\left \{any\;digit\;except\; d\right \}\rightarrow 9 choices
Choices for f=\left \{any\;digit\;except\; e\right \}\rightarrow 9 choices
Total count of six-digit numbers that do not have adjacent digits the same
9\times 9\times 9\times 9\times 9\times 9=531441

Answer: 531441

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About mvtrinh

Retired high school math teacher.
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