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Question Number 40284 by scientist last updated on 18/Jul/18
How many permutation can be made using the word  CANADA each of the six letter being used mixed  in each permutation?  In how many of these  permutation will the three A′s be together? In   how many will two A′s be together but not the three?
$${How}\:{many}\:{permutation}\:{can}\:{be}\:{made}\:{using}\:{the}\:{word} \\ $$$${CANADA}\:{each}\:{of}\:{the}\:{six}\:{letter}\:{being}\:{used}\:{mixed} \\ $$$${in}\:{each}\:{permutation}?\:\:{In}\:{how}\:{many}\:{of}\:{these} \\ $$$${permutation}\:{will}\:{the}\:{three}\:{A}'{s}\:{be}\:{together}?\:{In}\: \\ $$$${how}\:{many}\:{will}\:{two}\:{A}'{s}\:{be}\:{together}\:{but}\:{not}\:{the}\:{three}? \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 19/Jul/18
i)CANADA   A=3 C=1 D=1 N=1  permutaion numbers=((6!)/(3!))=((6×5×4×3!)/(3!))=120  ii)let 03 numbers of A faviquicked with each  other so (AAA)CDN   permutation=4!=24  iii)(AA)ACDN  pdrmutation=3C_2 ×5!=3×120=360  pls check...  after editing...  iii)required answer is 360−24=336  becos  out of these 360 permutation  24  permutation are included where 03  AAA will  be together...so ans is 360−24=336
$$\left.{i}\right){CANADA}\:\:\:{A}=\mathrm{3}\:{C}=\mathrm{1}\:{D}=\mathrm{1}\:{N}=\mathrm{1} \\ $$$${permutaion}\:{numbers}=\frac{\mathrm{6}!}{\mathrm{3}!}=\frac{\mathrm{6}×\mathrm{5}×\mathrm{4}×\mathrm{3}!}{\mathrm{3}!}=\mathrm{120} \\ $$$$\left.{ii}\right){let}\:\mathrm{03}\:{numbers}\:{of}\:{A}\:{faviquicked}\:{with}\:{each} \\ $$$${other}\:{so}\:\left({AAA}\right){CDN}\: \\ $$$${permutation}=\mathrm{4}!=\mathrm{24} \\ $$$$\left.{iii}\right)\left({AA}\right){ACDN} \\ $$$${pdrmutation}=\mathrm{3}{C}_{\mathrm{2}} ×\mathrm{5}!=\mathrm{3}×\mathrm{120}=\mathrm{360} \\ $$$${pls}\:{check}… \\ $$$${after}\:{editing}… \\ $$$$\left.{iii}\right){required}\:{answer}\:{is}\:\mathrm{360}−\mathrm{24}=\mathrm{336} \\ $$$${becos}\:\:{out}\:{of}\:{these}\:\mathrm{360}\:{permutation}\:\:\mathrm{24} \\ $$$${permutation}\:{are}\:{included}\:{where}\:\mathrm{03}\:\:{AAA}\:{will} \\ $$$${be}\:{together}…{so}\:{ans}\:{is}\:\mathrm{360}−\mathrm{24}=\mathrm{336} \\ $$
Commented by MJS last updated on 18/Jul/18
iii) might be wrong because AAA is not  allowed but I′m not sure in these things
$$\left.{iii}\right)\:\mathrm{might}\:\mathrm{be}\:\mathrm{wrong}\:\mathrm{because}\:{AAA}\:\mathrm{is}\:\mathrm{not} \\ $$$$\mathrm{allowed}\:\mathrm{but}\:\mathrm{I}'\mathrm{m}\:\mathrm{not}\:\mathrm{sure}\:\mathrm{in}\:\mathrm{these}\:\mathrm{things} \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 19/Jul/18
yes you are right...i am editing...
$${yes}\:{you}\:{are}\:{right}…{i}\:{am}\:{editing}… \\ $$

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