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Category: Permutation and Combination

How-many-bit-strings-of-length-11-have-exactly-three-consecutive-1s-

Question Number 203833 by depressiveshrek last updated on 29/Jan/24 $$\mathrm{How}\:\mathrm{many}\:\mathrm{bit}\:\mathrm{strings}\:\mathrm{of}\:\mathrm{length}\:\mathrm{11}\:\mathrm{have} \\ $$$$\mathrm{exactly}\:\mathrm{three}\:\mathrm{consecutive}\:\mathrm{1s}? \\ $$ Answered by nikif99 last updated on 30/Jan/24 $${Such}\:{a}\:{typical}\:{bit}\:{string}\:{of}\:{length}=\mathrm{11}\:{is} \\ $$$$\mathrm{000}\underset{{pos}=\mathrm{4}} {\underbrace{\mathrm{111}}00000}…

How-many-bit-strings-of-length-10-do-not-have-four-consecutive-1s-

Question Number 203832 by depressiveshrek last updated on 29/Jan/24 $$\mathrm{How}\:\mathrm{many}\:\mathrm{bit}\:\mathrm{strings}\:\mathrm{of}\:\mathrm{length}\:\mathrm{10}\:\mathrm{do}\:\mathrm{not} \\ $$$$\mathrm{have}\:\mathrm{four}\:\mathrm{consecutive}\:\mathrm{1s}? \\ $$ Answered by nikif99 last updated on 30/Jan/24 $${Total}\:{number}\:{of}\:{bit}\:{strings}\:{of}\:{length}\:\mathrm{10} \\ $$$${is}\:\mathrm{2}^{\mathrm{10}} =\mathrm{1024}\:\left(\mathrm{0}−\mathrm{1023}\right).…