6-people-a-b-c-d-e-and-f-stand-in-a-line-The-number-of-ways-they-can-stand-arranged-is-equal-to-6-If-two-people-have-to-stand-next-to-each-other-but-everyone-else-do-not-matter-how-many-c

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Question Number 10387 by FilupSmith last updated on 06/Feb/17

$$6\mathrm{people}a,b,c,d,e,\mathrm{and}f\mathrm{stand}\mathrm{in}\mathrm{a}\mathrm{line}.$$$$$$$$\mathrm{The}\mathrm{number}\mathrm{of}\mathrm{ways}\mathrm{they}\mathrm{can}\mathrm{stand}\mathrm{arranged}$$$$\mathrm{is}\mathrm{equal}\mathrm{to}6!$$$$$$$$\mathrm{If}\mathrm{two}\mathrm{people}\mathrm{have}\mathrm{to}\mathrm{stand}\mathrm{next}\mathrm{to}\mathrm{each}\mathrm{other},$$$$\mathrm{but}\mathrm{everyone}\mathrm{else}\mathrm{do}\mathrm{not}\mathrm{matter},\mathrm{how}\mathrm{many}\mathrm{combinations}$$$$\mathrm{combinations}\mathrm{are}\mathrm{there}?$$$$$$$$\mathrm{e}.\mathrm{g}.\left(ab\right)cdef\mathrm{or}cd\left(ba\right)ef$$

Answered by sandy_suhendra last updated on 06/Feb/17

$$\mathrm{the}\mathrm{two}\mathrm{people}\mathrm{we}\mathrm{assume}\mathrm{as}1\mathrm{people}$$$$\mathrm{so}\mathrm{there}\mathrm{are}5\mathrm{people}\Rightarrow \mathrm{the}\mathrm{combination}=5!$$$$\mathrm{but}\mathrm{the}\mathrm{two}\mathrm{people}\mathrm{have}\mathrm{combination}=2!$$$$\mathrm{the}\mathrm{total}\mathrm{combination}=2!\times 5!=240$$

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