There-are-12-students-in-a-party-Five-of-them-are-girls-In-how-many-ways-can-these-12-students-be-arranged-in-a-row-if-i-there-are-no-restrictions-ii-the-5-girls-must-be-together-forming-a-b

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Question Number 125208 by liberty last updated on 09/Dec/20

$$Thereare12studentsinaparty.Fiveof$$$$themaregirls.Inhowmanywayscan$$$$these12studentsbearrangedinarowif$$$$\left(i\right)therearenorestrictions?$$$$\left(ii\right)the5girlsmustbetogether\left(formingablock\right)?$$$$\left(iii\right)no2girlsareadjacent$$$$\left(iv\right)betweentwoparticularboysAandB$$$$,therenoboysbutexactly3girls?$$

Answered by john_santu last updated on 09/Dec/20

$$\left(i\right)12!$$$$\left(ii\right)5!(7+1)!=5!8!$$$$\left(iii\right)\underset{1}{-}{G}_1\underset{2}{\stackrel{x}{-}}{G}_2\underset{3}{\stackrel{x}{-}}{G}_3\underset{4}{\stackrel{x}{-}}{G}_4\underset{5}{\stackrel{x}{-}}{G}_5\underset{6}{-}$$$$=5!7!\left(\begin{array}{c}6+3-1\\ 3\end{array}\right)=5!7!\left(\begin{array}{c}8\\ 3\end{array}\right)$$$$\left(iv\right)AGGGB-------$$$$-AGGGB------$$$$--AGGGB-----$$$$---AGGGB----$$$$----AGGGB---$$$$-----AGGGB--$$$$------AGGGB-$$$$-------AGGGB$$$$thenumberofwaysis$$$$=2\times 8\times C_3^5\times 3!\times 7!$$$$=12\times 8!\times 10=120\times 8!$$

Commented by liberty last updated on 09/Dec/20

$$yes..correct..thanks$$

Commented by malwan last updated on 16/Jan/21

$$please$$$$I{can}^{\prime }tunderstandnumber\left(iii\right)$$$$\left(\begin{array}{c}6+3-1\\ 3\end{array}\right)whatisit?6?3?$$$$ifgirls=3andboys=6then$$$$whatisthesolution?$$$$andwhatifgirls=nand$$$$boys=m;m>n?$$

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