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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).…

There-are-many-ways-to-arrange-3-red-balls-and-9-black-balls-in-a-circle-so-that-there-are-a-minimum-of-2-black-balls-between-2-adjacent-red-balls-a-180-8-b-240-7-c-364-6-d-282-4

Question Number 200051 by cortano12 last updated on 13/Nov/23 $$ \\ $$$$\mathrm{There}\:\mathrm{are}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{to}\:\mathrm{arrange}\:\mathrm{3}\:\mathrm{red} \\ $$$$\:\mathrm{balls}\:\mathrm{and}\:\mathrm{9}\:\mathrm{black}\:\mathrm{balls}\:\mathrm{in}\:\mathrm{a}\:\mathrm{circle}\: \\ $$$$\mathrm{so}\:\mathrm{that}\:\mathrm{there}\:\mathrm{are}\:\mathrm{a}\:\mathrm{minimum}\:\mathrm{of}\:\mathrm{2} \\ $$$$\mathrm{black}\:\mathrm{balls}\:\mathrm{between}\:\mathrm{2}\:\mathrm{adjacent}\:\mathrm{red} \\ $$$$\mathrm{balls}. \\ $$$$\left(\mathrm{a}\right)\:\mathrm{180}×\mathrm{8}!\:\:\:\:\left(\mathrm{b}\right)\:\mathrm{240}×\mathrm{7}!\:\:\:\:\left(\mathrm{c}\right)\:\mathrm{364}×\mathrm{6}! \\ $$$$\:\left(\mathrm{d}\right)\:\mathrm{282}×\mathrm{4}!\:\:\:\:\left(\mathrm{e}\right)\:\mathrm{144}×\mathrm{5}!\: \\…