Question Number 54698 by mr W last updated on 09/Feb/19
$${How}\:{many}\:{words}\:{with}\:{at}\:{least}\:\mathrm{2}\:{letters} \\ $$$${can}\:{be}\:{formed}\:{using}\:{the}\:{letters}\:{from} \\ $$$${TINKUTARA}? \\ $$
Answered by afachri last updated on 09/Feb/19
Commented by mr W last updated on 10/Feb/19
$${thank}\:{you}\:{sir}! \\ $$$${please}\:{recheck}\:{your}\:{results}. \\ $$$${for}\:{example}\:{words}\:{with}\:\mathrm{2}\:{letters}: \\ $$$$ \\ $$$${with}\:\mathrm{2}\:{same}\:{letters}:\:\mathrm{2}\:{ways},\:{i}.{e}.\:{TT}\:{and}\:{AA}. \\ $$$$ \\ $$$${with}\:\mathrm{2}\:{different}\:{letters}:\:{we}\:{can}\:{choose} \\ $$$$\mathrm{2}\:{letters}\:{from}\:\mathrm{7}\:{letters}\:\left({i}.{e}.\:{TINKUAR}\right), \\ $$$${and}\:{we}\:{have}\:\mathrm{7}×\mathrm{6}=\mathrm{42}\:{ways}. \\ $$$$ \\ $$$${that}\:{is}\:{to}\:{say}\:{there}\:{are}\:\mathrm{2}+\mathrm{42}=\mathrm{44}\:{ways} \\ $$$${to}\:{form}\:{words}\:{with}\:\mathrm{2}\:{letters},\:{not}\:\mathrm{18}\:{ways}. \\ $$
Commented by mr W last updated on 10/Feb/19
$${case}\:“{words}\:{with}\:\mathrm{3}\:{letters}'': \\ $$$${words}\:{with}\:\mathrm{2}\:{letters}\:{TT}:\:{TTx},\:{TxT},\:{xTT}\Rightarrow\mathrm{3}×\mathrm{6}=\mathrm{18}\:{ways} \\ $$$${words}\:{with}\:\mathrm{2}\:{letters}\:{AA}:\:{AAx},\:{AxA},\:{xAA}\Rightarrow\mathrm{3}×\mathrm{6}=\mathrm{18}\:{ways} \\ $$$${words}\:{with}\:\mathrm{3}\:{different}\:{letters}:\:\mathrm{7}×\mathrm{6}×\mathrm{5}=\mathrm{210}\:{ways} \\ $$$$\Rightarrow{totally}\:\mathrm{2}×\mathrm{18}+\mathrm{210}=\mathrm{246}\:{ways},\:{not}\:\mathrm{126}\:{ways}. \\ $$
Commented by mr W last updated on 10/Feb/19
$${case}\:“{words}\:{with}\:\mathrm{4}\:{letters}'': \\ $$$${words}\:{with}\:\mathrm{2}\:{letters}\:{TT}\:{and}\:\mathrm{2}\:{letters}\:{AA}: \\ $$$$\frac{\mathrm{4}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{6} \\ $$$$ \\ $$$${words}\:{with}\:\mathrm{2}\:{letters}\:{TT},\:{other}\:{letters}\:{are}\:{different}: \\ $$$${C}_{\mathrm{2}} ^{\mathrm{6}} ×\frac{\mathrm{4}!}{\mathrm{2}!}=\mathrm{180} \\ $$$$ \\ $$$${words}\:{with}\:\mathrm{2}\:{letters}\:{AA},\:{other}\:{letters}\:{are}\:{different}: \\ $$$${C}_{\mathrm{2}} ^{\mathrm{6}} ×\frac{\mathrm{4}!}{\mathrm{2}!}=\mathrm{180} \\ $$$$ \\ $$$${words}\:{with}\:{different}\:{letters}: \\ $$$$\mathrm{7}×\mathrm{6}×\mathrm{5}×\mathrm{4}=\mathrm{840} \\ $$$$ \\ $$$$\Rightarrow{totally}\:\mathrm{6}+\mathrm{180}×\mathrm{2}+\mathrm{840}=\mathrm{1206}\:{ways} \\ $$
Commented by afachri last updated on 10/Feb/19
$$\boldsymbol{\mathrm{aahh}}\:\boldsymbol{\mathrm{i}}\:\boldsymbol{\mathrm{did}}\:\boldsymbol{\mathrm{it}}\:\boldsymbol{\mathrm{wrong}}.\:\boldsymbol{\mathrm{thank}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{illuminate}} \\ $$$$\boldsymbol{\mathrm{me}}\:\boldsymbol{\mathrm{Mr}}\:\boldsymbol{\mathrm{W}}. \\ $$
Commented by afachri last updated on 10/Feb/19
$$\boldsymbol{\mathrm{so}}\:\boldsymbol{{S}}{ir},\:{why}\:{the}\:\mathrm{5}\:{letters}\:{case}\:{is}\:{not} \\ $$$${counted}\:?? \\ $$
Commented by mr W last updated on 10/Feb/19
$${i}\:{have}\:{not}\:{checked}\:{the}\:{other}\:{cases}. \\ $$
Commented by afachri last updated on 11/Feb/19
$${yes}\:{Sir} \\ $$
Answered by mr W last updated on 12/Feb/19
$${we}\:{have}\:{totally}\:\mathrm{9}\:{letters}: \\ $$$${TT} \\ $$$${AA} \\ $$$${INKUR} \\ $$$$ \\ $$$$−\:{words}\:{with}\:\mathrm{2}\:{letters} \\ $$$${with}\:{same}\:{letters}:\:\mathrm{2}\:{ways},\:{i}.{e}.\:{TT}\:{and}\:{AA} \\ $$$${with}\:{different}\:{letters}:\:{P}_{\mathrm{2}} ^{\mathrm{7}} =\mathrm{7}×\mathrm{6}=\mathrm{42}\:{ways}\: \\ $$$$\Rightarrow\mathrm{2}+\mathrm{42}=\mathrm{44}\:{ways} \\ $$$$ \\ $$$$−\:{words}\:{with}\:\mathrm{3}\:{letters} \\ $$$${with}\:\mathrm{2}\:{same}\:{letters}\:{TT}:\:{C}_{\mathrm{1}} ^{\mathrm{6}} ×\frac{\mathrm{3}!}{\mathrm{2}!}=\mathrm{18}\:{ways} \\ $$$${with}\:\mathrm{2}\:{same}\:{letters}\:{AA}:\:{C}_{\mathrm{1}} ^{\mathrm{6}} ×\frac{\mathrm{3}!}{\mathrm{2}!}=\mathrm{18}\:{ways} \\ $$$${with}\:{different}\:{letters}:\:{P}_{\mathrm{3}} ^{\mathrm{7}} =\mathrm{7}×\mathrm{6}×\mathrm{5}=\mathrm{210}\:{ways} \\ $$$$\Rightarrow\mathrm{2}×\mathrm{18}+\mathrm{210}=\mathrm{246}\:{ways} \\ $$$$ \\ $$$$−\:{words}\:{with}\:\mathrm{4}\:{letters} \\ $$$${with}\:\mathrm{2}\:{TT}\:{and}\:\mathrm{2}\:{AA}:\:\frac{\mathrm{4}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{6}\:{ways} \\ $$$${with}\:\mathrm{2}\:{TT}\:{and}\:\mathrm{2}\:{other}\:{different}\:{letters}:\:{C}_{\mathrm{2}} ^{\mathrm{6}} ×\frac{\mathrm{4}!}{\mathrm{2}!}=\mathrm{180}\:{ways} \\ $$$${with}\:\mathrm{2}\:{AA}\:{and}\:\mathrm{2}\:{other}\:{different}\:{letters}:\:{C}_{\mathrm{2}} ^{\mathrm{6}} ×\frac{\mathrm{4}!}{\mathrm{2}!}=\mathrm{180}\:{ways} \\ $$$${with}\:\mathrm{4}\:{different}\:{letters}:\:{P}_{\mathrm{4}} ^{\mathrm{7}} =\mathrm{7}×\mathrm{6}×\mathrm{5}×\mathrm{4}=\mathrm{840}\:{ways} \\ $$$$\Rightarrow\mathrm{6}+\mathrm{2}×\mathrm{180}+\mathrm{840}=\mathrm{1206}\:{ways} \\ $$$$ \\ $$$$−\:{words}\:{with}\:\mathrm{5}\:{letters} \\ $$$${TT}+{AA}+{x}:\:{C}_{\mathrm{1}} ^{\mathrm{5}} ×\frac{\mathrm{5}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{150}\:{ways} \\ $$$${TT}+{xxx}:\:{C}_{\mathrm{3}} ^{\mathrm{6}} ×\frac{\mathrm{5}!}{\mathrm{2}!}=\mathrm{1200}\:{ways} \\ $$$${AA}+{xxx}:\:{C}_{\mathrm{3}} ^{\mathrm{6}} ×\frac{\mathrm{5}!}{\mathrm{2}!}=\mathrm{1200}\:{ways} \\ $$$${with}\:\mathrm{5}\:{different}\:{letters}:\:{P}_{\mathrm{5}} ^{\mathrm{7}} =\mathrm{2520}\:{ways} \\ $$$$\Rightarrow\mathrm{150}+\mathrm{2}×\mathrm{1200}+\mathrm{2520}=\mathrm{5070}\:{ways} \\ $$$$ \\ $$$$−\:{words}\:{with}\:\mathrm{6}\:{letters} \\ $$$${TT}+{AA}+{xx}:\:{C}_{\mathrm{2}} ^{\mathrm{5}} ×\frac{\mathrm{6}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{1800}\:{ways} \\ $$$${TT}+{xxxx}:\:{C}_{\mathrm{4}} ^{\mathrm{6}} ×\frac{\mathrm{6}!}{\mathrm{2}!}=\mathrm{5400}\:{ways} \\ $$$${AA}+{xxxx}:\:{C}_{\mathrm{4}} ^{\mathrm{6}} ×\frac{\mathrm{6}!}{\mathrm{2}!}=\mathrm{5400}\:{ways} \\ $$$${with}\:\mathrm{6}\:{different}\:{letters}:\:{P}_{\mathrm{6}} ^{\mathrm{7}} =\mathrm{5040}\:{ways} \\ $$$$\Rightarrow\mathrm{1800}+\mathrm{2}×\mathrm{5400}+\mathrm{5040}=\mathrm{17640}\:{ways} \\ $$$$ \\ $$$$−\:{words}\:{with}\:\mathrm{7}\:{letters} \\ $$$${TT}+{AA}+{xxx}:\:{C}_{\mathrm{3}} ^{\mathrm{5}} ×\frac{\mathrm{7}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{12600}\:{ways} \\ $$$${TT}+{xxxxx}:\:{C}_{\mathrm{5}} ^{\mathrm{6}} ×\frac{\mathrm{7}!}{\mathrm{2}!}=\mathrm{15120}\:{ways} \\ $$$${AA}+{xxxxx}:\:{C}_{\mathrm{5}} ^{\mathrm{6}} ×\frac{\mathrm{7}!}{\mathrm{2}!}=\mathrm{15120}\:{ways} \\ $$$${with}\:\mathrm{7}\:{different}\:{letters}:\:{P}_{\mathrm{7}} ^{\mathrm{7}} =\mathrm{5040}\:{ways} \\ $$$$\Rightarrow\mathrm{12600}+\mathrm{2}×\mathrm{15120}+\mathrm{5040}=\mathrm{47880}\:{ways} \\ $$$$ \\ $$$$−\:{words}\:{with}\:\mathrm{8}\:{letters} \\ $$$${TT}+{AA}+{xxxx}:\:{C}_{\mathrm{4}} ^{\mathrm{5}} ×\frac{\mathrm{8}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{50400}\:{ways} \\ $$$${TT}+{xxxxxx}:\:{C}_{\mathrm{6}} ^{\mathrm{6}} ×\frac{\mathrm{8}!}{\mathrm{2}!}=\mathrm{20160}\:{ways} \\ $$$${AA}+{xxxxxx}:\:{C}_{\mathrm{6}} ^{\mathrm{6}} ×\frac{\mathrm{8}!}{\mathrm{2}!}=\mathrm{20160}\:{ways} \\ $$$$\Rightarrow\mathrm{50400}+\mathrm{2}×\mathrm{20160}=\mathrm{90720} \\ $$$$ \\ $$$$−\:{words}\:{with}\:\mathrm{9}\:{letters} \\ $$$$\Rightarrow\frac{\mathrm{9}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{90720} \\ $$$$ \\ $$$${sum}:\:\mathrm{44}+\mathrm{246}+\mathrm{1206}+\mathrm{5070}+\mathrm{17640}+\mathrm{47880}+\mathrm{90720}+\mathrm{90720} \\ $$$$=\mathrm{253526}\:{ways} \\ $$