Question and Answers Forum

All Questions      Topic List

Permutation and Combination Questions

Previous in All Question      Next in All Question      

Previous in Permutation and Combination      Next in Permutation and Combination      

Question Number 21760 by Tinkutara last updated on 03/Oct/17

There are m points on the line AB and n points on the line AC, excluding the point A. Triangles are formed joining these points (i) When point A is not included, (ii) When point A is included. The ratio of the number of such triangles is

$$\mathrm{There}\:\mathrm{are}\:{m}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the}\:\mathrm{line}\:{AB}\:\mathrm{and} \\ $$$${n}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the}\:\mathrm{line}\:{AC},\:\mathrm{excluding}\:\mathrm{the} \\ $$$$\mathrm{point}\:{A}.\:\mathrm{Triangles}\:\mathrm{are}\:\mathrm{formed}\:\mathrm{joining} \\ $$$$\mathrm{these}\:\mathrm{points} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{When}\:\mathrm{point}\:{A}\:\mathrm{is}\:\mathrm{not}\:\mathrm{included}, \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{When}\:\mathrm{point}\:{A}\:\mathrm{is}\:\mathrm{included}. \\ $$$$\mathrm{The}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{such}\:\mathrm{triangles} \\ $$$$\mathrm{is} \\ $$

Answered by mrW1 last updated on 03/Oct/17

(i) mC_2 ^n +nC_2 ^m =((mn(m+n−2))/2) (ii) mC_2 ^n +nC_2 ^m +mn=((mn(m+n−2))/2)+mn=((mn(m+n))/2) ratio =((m+n−2)/(m+n))

$$\left(\mathrm{i}\right) \\ $$$$\mathrm{mC}_{\mathrm{2}} ^{\mathrm{n}} +\mathrm{nC}_{\mathrm{2}} ^{\mathrm{m}} =\frac{\mathrm{mn}\left(\mathrm{m}+\mathrm{n}−\mathrm{2}\right)}{\mathrm{2}} \\ $$$$\left(\mathrm{ii}\right) \\ $$$$\mathrm{mC}_{\mathrm{2}} ^{\mathrm{n}} +\mathrm{nC}_{\mathrm{2}} ^{\mathrm{m}} +\mathrm{mn}=\frac{\mathrm{mn}\left(\mathrm{m}+\mathrm{n}−\mathrm{2}\right)}{\mathrm{2}}+\mathrm{mn}=\frac{\mathrm{mn}\left(\mathrm{m}+\mathrm{n}\right)}{\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{ratio}\:=\frac{\mathrm{m}+\mathrm{n}−\mathrm{2}}{\mathrm{m}+\mathrm{n}} \\ $$

Commented by Tinkutara last updated on 03/Oct/17

Thank you very much Sir!

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$

Commented by mrW1 last updated on 03/Oct/17

there are mn triangles with point A. you can also calculate the number of triangles in (ii) like this: mC_2 ^(n+1) +nC_2 ^m =((m(n+1)n)/2)+((nm(m−1))/2)=((mn(m+n))/2)

$$\mathrm{there}\:\mathrm{are}\:\mathrm{mn}\:\mathrm{triangles}\:\mathrm{with}\:\mathrm{point}\:\mathrm{A}. \\ $$$$ \\ $$$$\mathrm{you}\:\mathrm{can}\:\mathrm{also}\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{triangles}\:\mathrm{in}\:\left(\mathrm{ii}\right)\:\mathrm{like}\:\mathrm{this}: \\ $$$$\mathrm{mC}_{\mathrm{2}} ^{\mathrm{n}+\mathrm{1}} +\mathrm{nC}_{\mathrm{2}} ^{\mathrm{m}} =\frac{\mathrm{m}\left(\mathrm{n}+\mathrm{1}\right)\mathrm{n}}{\mathrm{2}}+\frac{\mathrm{nm}\left(\mathrm{m}−\mathrm{1}\right)}{\mathrm{2}}=\frac{\mathrm{mn}\left(\mathrm{m}+\mathrm{n}\right)}{\mathrm{2}} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com