Question and Answers Forum

All Questions      Topic List

Probability and Statistics Questions

Previous in All Question      Next in All Question      

Previous in Probability and Statistics      Next in Probability and Statistics      

Question Number 133424 by liberty last updated on 22/Feb/21

Given 10 white balls and ten black balls  numbered 1,2,...,10. How many   ways can we choose 6 balls such that  (i) no two chosen balls have the same number  (ii) two pairs of chosen balls have  the same number?

$$\mathrm{Given}\:\mathrm{10}\:\mathrm{white}\:\mathrm{balls}\:\mathrm{and}\:\mathrm{ten}\:\mathrm{black}\:\mathrm{balls} \\ $$$$\mathrm{numbered}\:\mathrm{1},\mathrm{2},...,\mathrm{10}.\:\mathrm{How}\:\mathrm{many}\: \\ $$$$\mathrm{ways}\:\mathrm{can}\:\mathrm{we}\:\mathrm{choose}\:\mathrm{6}\:\mathrm{balls}\:\mathrm{such}\:\mathrm{that} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{no}\:\mathrm{two}\:\mathrm{chosen}\:\mathrm{balls}\:\mathrm{have}\:\mathrm{the}\:\mathrm{same}\:\mathrm{number} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{two}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{chosen}\:\mathrm{balls}\:\mathrm{have} \\ $$$$\mathrm{the}\:\mathrm{same}\:\mathrm{number}? \\ $$

Answered by EDWIN88 last updated on 22/Feb/21

(i) six number from the set {1,2,...,10} can be  chosen  (((10)),((  6)) ) ways. Each of these numbers  can appear in two ways on the chosen balls.  There are  (((10)),((  6)) ). 2^6  = 13 440 ways of choosing  6 balls such that no two of them are denoted   by the same number  (ii) two pairs of balls having the same number can  be chosen  (((10)),((  2)) ) ways. From the remaining  16 balls one can choose two balls numbered  differently in  ((8),(2) ) .2^2  ways. The answer in  this case is  (((10)),((  2)) )  ((8),(2) ) .2^2  = 5 040

$$\left(\mathrm{i}\right)\:\mathrm{six}\:\mathrm{number}\:\mathrm{from}\:\mathrm{the}\:\mathrm{set}\:\left\{\mathrm{1},\mathrm{2},...,\mathrm{10}\right\}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{chosen}\:\begin{pmatrix}{\mathrm{10}}\\{\:\:\mathrm{6}}\end{pmatrix}\:\mathrm{ways}.\:\mathrm{Each}\:\mathrm{of}\:\mathrm{these}\:\mathrm{numbers} \\ $$$$\mathrm{can}\:\mathrm{appear}\:\mathrm{in}\:\mathrm{two}\:\mathrm{ways}\:\mathrm{on}\:\mathrm{the}\:\mathrm{chosen}\:\mathrm{balls}. \\ $$$$\mathrm{There}\:\mathrm{are}\:\begin{pmatrix}{\mathrm{10}}\\{\:\:\mathrm{6}}\end{pmatrix}.\:\mathrm{2}^{\mathrm{6}} \:=\:\mathrm{13}\:\mathrm{440}\:\mathrm{ways}\:\mathrm{of}\:\mathrm{choosing} \\ $$$$\mathrm{6}\:\mathrm{balls}\:\mathrm{such}\:\mathrm{that}\:\mathrm{no}\:\mathrm{two}\:\mathrm{of}\:\mathrm{them}\:\mathrm{are}\:\mathrm{denoted}\: \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{same}\:\mathrm{number} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{two}\:\mathrm{pairs}\:\mathrm{of}\:\mathrm{balls}\:\mathrm{having}\:\mathrm{the}\:\mathrm{same}\:\mathrm{number}\:\mathrm{can} \\ $$$$\mathrm{be}\:\mathrm{chosen}\:\begin{pmatrix}{\mathrm{10}}\\{\:\:\mathrm{2}}\end{pmatrix}\:\mathrm{ways}.\:\mathrm{From}\:\mathrm{the}\:\mathrm{remaining} \\ $$$$\mathrm{16}\:\mathrm{balls}\:\mathrm{one}\:\mathrm{can}\:\mathrm{choose}\:\mathrm{two}\:\mathrm{balls}\:\mathrm{numbered} \\ $$$$\mathrm{differently}\:\mathrm{in}\:\begin{pmatrix}{\mathrm{8}}\\{\mathrm{2}}\end{pmatrix}\:.\mathrm{2}^{\mathrm{2}} \:\mathrm{ways}.\:\mathrm{The}\:\mathrm{answer}\:\mathrm{in} \\ $$$$\mathrm{this}\:\mathrm{case}\:\mathrm{is}\:\begin{pmatrix}{\mathrm{10}}\\{\:\:\mathrm{2}}\end{pmatrix}\:\begin{pmatrix}{\mathrm{8}}\\{\mathrm{2}}\end{pmatrix}\:.\mathrm{2}^{\mathrm{2}} \:=\:\mathrm{5}\:\mathrm{040}\: \\ $$$$ \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com