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If-four-people-all-said-HELLO-to-each-other-once-how-many-times-would-the-word-HELLO-be-spoken-

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Question Number 5897 by sanusihammed last updated on 04/Jun/16
If four people all said HELLO to each other once. how many times would the word HELLO be spoken ?
$${If}\:{four}\:{people}\:{all}\:{said}\:\:{HELLO}\:{to}\:{each}\:{other}\:{once}.\:{how}\:{many}\: \\ $$$${times}\:{would}\:{the}\:{word}\:\:{HELLO}\:\:{be}\:{spoken}\:? \\ $$
Answered by FilupSmith last updated on 04/Jun/16
person 1 talks to 3 people person 2 talks to 2 people person 3 talks to 1 people everyone has talked to person 4. ∴ = 3+2+1 times talked = 6
$${person}\:\mathrm{1}\:{talks}\:{to}\:\mathrm{3}\:{people} \\ $$$${person}\:\mathrm{2}\:{talks}\:{to}\:\mathrm{2}\:{people} \\ $$$${person}\:\mathrm{3}\:{talks}\:{to}\:\mathrm{1}\:{people} \\ $$$${everyone}\:{has}\:{talked}\:{to}\:{person}\:\mathrm{4}. \\ $$$$\therefore\:=\:\mathrm{3}+\mathrm{2}+\mathrm{1} \\ $$$${times}\:{talked}\:=\:\mathrm{6} \\ $$
Commented by FilupSmith last updated on 04/Jun/16
for n people: times talked = Σ_(i=1) ^(n−1) i =((n^2 −n)/2)
$${for}\:{n}\:{people}: \\ $$$${times}\:{talked}\:=\:\underset{{i}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\sum}}{i} \\ $$$$=\frac{{n}^{\mathrm{2}} −{n}}{\mathrm{2}} \\ $$
Commented by prakash jain last updated on 04/Jun/16
I think if two person say hello both of them will need to speak. so word hello will be spoken 6×2=12 times
$$\mathrm{I}\:\mathrm{think}\:\mathrm{if}\:\mathrm{two}\:\mathrm{person}\:\mathrm{say}\:\mathrm{hello}\:\mathrm{both}\:\mathrm{of}\:\mathrm{them} \\ $$$$\mathrm{will}\:\mathrm{need}\:\mathrm{to}\:\mathrm{speak}. \\ $$$$\mathrm{so}\:\mathrm{word}\:\mathrm{hello}\:\mathrm{will}\:\mathrm{be}\:\mathrm{spoken}\:\mathrm{6}×\mathrm{2}=\mathrm{12}\:\mathrm{times} \\ $$
Commented by FilupSmith last updated on 05/Jun/16
if everyone said hello to each other, but no two people say hello to each other more than once: (1) says hello to 2 3 and 4 (2) says hello to 3 and 4 (1 already seid hello) 3 says hello to 4 everyones talked to 4 so 4 does nothing =3+2+1=6 this is true only if they talk to each other once only.
$${if}\:{everyone}\:{said}\:{hello}\:{to}\:{each}\:{other}, \\ $$$${but}\:{no}\:{two}\:{people}\:{say}\:{hello}\:{to}\:{each}\:{other} \\ $$$${more}\:{than}\:{once}: \\ $$$$\left(\mathrm{1}\right)\:{says}\:{hello}\:{to}\:\mathrm{2}\:\mathrm{3}\:{and}\:\mathrm{4} \\ $$$$\left(\mathrm{2}\right)\:{says}\:{hello}\:{to}\:\mathrm{3}\:{and}\:\mathrm{4}\:\left(\mathrm{1}\:{already}\:{seid}\:{hello}\right) \\ $$$$\mathrm{3}\:\mathrm{says}\:\mathrm{hello}\:\mathrm{to}\:\mathrm{4} \\ $$$$\mathrm{everyones}\:\mathrm{talked}\:\mathrm{to}\:\mathrm{4}\:\mathrm{so}\:\mathrm{4}\:\mathrm{does}\:\mathrm{nothing} \\ $$$$=\mathrm{3}+\mathrm{2}+\mathrm{1}=\mathrm{6} \\ $$$$ \\ $$$$\mathrm{this}\:\mathrm{is}\:\mathrm{true}\:\mathrm{only}\:\mathrm{if}\:\mathrm{they}\:\mathrm{talk}\:\mathrm{to}\:\mathrm{each}\:\mathrm{other} \\ $$$$\mathrm{once}\:\mathrm{only}. \\ $$
Commented by FilupSmith last updated on 05/Jun/16
if 4 people said hello to each other without restriction, each person says hello (n−1) times for n people =n(n−1) =4(3) =12
$${if}\:\mathrm{4}\:{people}\:{said}\:{hello}\:{to}\:{each}\:{other} \\ $$$${without}\:{restriction},\:{each}\:{person} \\ $$$${says}\:{hello}\:\left({n}−\mathrm{1}\right)\:{times}\:{for}\:{n}\:{people} \\ $$$$={n}\left({n}−\mathrm{1}\right) \\ $$$$=\mathrm{4}\left(\mathrm{3}\right) \\ $$$$=\mathrm{12} \\ $$

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