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