How many 20-digit numbers can be formed from <mo fence="false" stretchy="false">{ 1 , 2

pablos28spainzd 2022-07-07 Answered
How many 20-digit numbers can be formed from { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } such that no 2 consecutive digit is both odd
I've noticed that the number of odd digit in the number must be less than 11. But i can't progress more than that.
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Answers (1)

Freddy Doyle
Answered 2022-07-08 Author has 20 answers
Step 1
Let x n be the number of n-digit numbers which can be formed from { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } such that no two consecutive digits are both odd.
We split x n into two parts: a n counts the numbers where the last digit is odd and b n counts the ones where the last digit is even. Then x n = a n + b n .
Step 2
Moreover we can easily check that a 1 = 5, b 1 = 4, a 2 = 4 5 = 20, b 2 = 9 4 = 36.
Is there any recurrences which involve a n , b n , a n + 1 and b n + 1 ?

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