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Q: How many bit strings are there of length eight?

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1024

210=1024

Half of them.

there are 128 (2 to the power of 7) bit strings of length 7

There are no zero-length strings that start with 1 bit or end with 2 bits. In a zero-length string, there are no bits at all.

1024 of them.

There are 210.

-- There are 256 bit strings of length 8 . -- There are 4 bit strings of length 2, and you've restricted 2 of the 8 bits to 1 of those 4 . -- So you've restricted the whole byte to 1/4 of its possible values = 64 of them.

You are essentially asking how many different bit strings of length 9 are there, as the first 3 bits are fixed. The answer is 2^9 = 512.

n+1 (counting the empty string)

its 2 raised to power 8 i.e 256

Three of the bits are already determined. That leaves 5 .25 = 32

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