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How-many-permutations-of-the-letters-of-the-word-EINSTEIN-are-possible-if-the-EIN-groups-must-not-be-next-to-eachother-




Question Number 166787 by nadovic last updated on 27/Feb/22
How many permutations of the  letters of the word EINSTEIN are  possible if the EIN groups must not  be next to eachother?
$$\mathrm{How}\:\mathrm{many}\:\mathrm{permutations}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{letters}\:\mathrm{of}\:\mathrm{the}\:\mathrm{word}\:\mathrm{EINSTEIN}\:\mathrm{are} \\ $$$$\mathrm{possible}\:\mathrm{if}\:\mathrm{the}\:\mathrm{EIN}\:\mathrm{groups}\:\mathrm{must}\:\mathrm{not} \\ $$$$\mathrm{be}\:\mathrm{next}\:\mathrm{to}\:\mathrm{eachother}? \\ $$
Commented by mr W last updated on 28/Feb/22
do you mean “there must be two  EIN groups, but they must not be  next to each other”? then there are  only following 6 possibilities:  EINSTEIN  EINTSEIN  SEINTEIN  TEINSEIN  EINSEINT  EINTEINS
$${do}\:{you}\:{mean}\:“{there}\:{must}\:{be}\:{two} \\ $$$${EIN}\:{groups},\:{but}\:{they}\:{must}\:{not}\:{be} \\ $$$${next}\:{to}\:{each}\:{other}''?\:{then}\:{there}\:{are} \\ $$$${only}\:{following}\:\mathrm{6}\:{possibilities}: \\ $$$${EINSTEIN} \\ $$$${EINTSEIN} \\ $$$${SEINTEIN} \\ $$$${TEINSEIN} \\ $$$${EINSEINT} \\ $$$${EINTEINS} \\ $$
Commented by nadovic last updated on 28/Feb/22
I think you did not include the   different ways in which the letters  in the EIN groups can also be   arranged differently within   themselves. Like ENISTNIE,  without still bringing the groups  next to eachother. That will give  more than 6 ways.
$$\mathrm{I}\:\mathrm{think}\:\mathrm{you}\:\mathrm{did}\:\mathrm{not}\:\mathrm{include}\:\mathrm{the}\: \\ $$$$\mathrm{different}\:\mathrm{ways}\:\mathrm{in}\:\mathrm{which}\:\mathrm{the}\:\mathrm{letters} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{EIN}\:\mathrm{groups}\:\mathrm{can}\:\mathrm{also}\:\mathrm{be}\: \\ $$$$\mathrm{arranged}\:\mathrm{differently}\:\mathrm{within}\: \\ $$$$\mathrm{themselves}.\:{Like}\:{ENISTNIE}, \\ $$$${without}\:{still}\:{bringing}\:{the}\:{groups} \\ $$$${next}\:{to}\:{eachother}.\:\mathrm{That}\:\mathrm{will}\:\mathrm{give} \\ $$$$\mathrm{more}\:\mathrm{than}\:\mathrm{6}\:\mathrm{ways}. \\ $$$$ \\ $$$$ \\ $$$$ \\ $$
Commented by mr W last updated on 28/Feb/22
you didn′t make it clear in the  question what you exactly mean with   “EIN groups”!  when the question meant that what  you just said, then the answer is  6×(3!)^2 =216.
$${you}\:{didn}'{t}\:{make}\:{it}\:{clear}\:{in}\:{the} \\ $$$${question}\:{what}\:{you}\:{exactly}\:{mean}\:{with}\: \\ $$$$“{EIN}\:{groups}''! \\ $$$${when}\:{the}\:{question}\:{meant}\:{that}\:{what} \\ $$$${you}\:{just}\:{said},\:{then}\:{the}\:{answer}\:{is} \\ $$$$\mathrm{6}×\left(\mathrm{3}!\right)^{\mathrm{2}} =\mathrm{216}. \\ $$

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