How many different ten-digit number can be created using only 2s and 0s which is divisible by 4 and doesn't contain two neighbouring 0s?
I know that the last two digits as a number have to be a multiple of 4 so the number is: .
And how can I count the numbers not containing neighbouring 0s?
I assume that 0202020220 will count as such a ten-digit number even though it begins with a zero.
is the answer.
In the first eight places in the number we can have anywhere from no 0’s(k=0) to at most four 0’s(k=4).
Then there will be 8-k 2’s creating 9-k to place and separate the 0’s.
Hello, James!
I assume that the leading digit cannot be zero.How many different ten-digit number can be created using only 2s and 0s
which is divisible by 4 and doesn't contain two neighbouring 0s?
I know that the last two digits as a number have to be a multiple of 4
. . so the number ends in: .
And how can I count the numbers not containing neighbouring 0s?
So we have: .
. . Hence, we want 7-digit numbers with non-adjacent 0s.
I couldn't find a way to think through this task, so I drew a tree diagram.
. . And here's what I found . . .
. .
Hence, there are 34 seven-digit numbers with non-adjacent 0s.
Therefore, the answer is: .
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Plato allowed a leading zero, so his result is the next Fibonacci number, 55.
I love your solution, Plato!