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Question Number 54698 by mr W last updated on 09/Feb/19

How many words with at least 2 letters can be formed using the letters from TINKUTARA?

$${How}\:{many}\:{words}\:{with}\:{at}\:{least}\:\mathrm{2}\:{letters} \\ $$$${can}\:{be}\:{formed}\:{using}\:{the}\:{letters}\:{from} \\ $$$${TINKUTARA}? \\ $$

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 2 letters: with 2 same letters: 2 ways, i.e. TT and AA. with 2 different letters: we can choose 2 letters from 7 letters (i.e. TINKUAR), and we have 7×6=42 ways. that is to say there are 2+42=44 ways to form words with 2 letters, not 18 ways.

$${thank}\:{you}\:{sir}! \\ $$$${please}\:{recheck}\:{your}\:{results}. \\ $$$${for}\:{example}\:{words}\:{with}\:\mathrm{2}\:{letters}: \\ $$$$ \\ $$$${with}\:\mathrm{2}\:{same}\:{letters}:\:\mathrm{2}\:{ways},\:{i}.{e}.\:{TT}\:{and}\:{AA}. \\ $$$$ \\ $$$${with}\:\mathrm{2}\:{different}\:{letters}:\:{we}\:{can}\:{choose} \\ $$$$\mathrm{2}\:{letters}\:{from}\:\mathrm{7}\:{letters}\:\left({i}.{e}.\:{TINKUAR}\right), \\ $$$${and}\:{we}\:{have}\:\mathrm{7}×\mathrm{6}=\mathrm{42}\:{ways}. \\ $$$$ \\ $$$${that}\:{is}\:{to}\:{say}\:{there}\:{are}\:\mathrm{2}+\mathrm{42}=\mathrm{44}\:{ways} \\ $$$${to}\:{form}\:{words}\:{with}\:\mathrm{2}\:{letters},\:{not}\:\mathrm{18}\:{ways}. \\ $$

Commented by mr W last updated on 10/Feb/19

case “words with 3 letters”: words with 2 letters TT: TTx, TxT, xTT⇒3×6=18 ways words with 2 letters AA: AAx, AxA, xAA⇒3×6=18 ways words with 3 different letters: 7×6×5=210 ways ⇒totally 2×18+210=246 ways, not 126 ways.

$${case}\:``{words}\:{with}\:\mathrm{3}\:{letters}'': \\ $$$${words}\:{with}\:\mathrm{2}\:{letters}\:{TT}:\:{TTx},\:{TxT},\:{xTT}\Rightarrow\mathrm{3}×\mathrm{6}=\mathrm{18}\:{ways} \\ $$$${words}\:{with}\:\mathrm{2}\:{letters}\:{AA}:\:{AAx},\:{AxA},\:{xAA}\Rightarrow\mathrm{3}×\mathrm{6}=\mathrm{18}\:{ways} \\ $$$${words}\:{with}\:\mathrm{3}\:{different}\:{letters}:\:\mathrm{7}×\mathrm{6}×\mathrm{5}=\mathrm{210}\:{ways} \\ $$$$\Rightarrow{totally}\:\mathrm{2}×\mathrm{18}+\mathrm{210}=\mathrm{246}\:{ways},\:{not}\:\mathrm{126}\:{ways}. \\ $$

Commented by mr W last updated on 10/Feb/19

case “words with 4 letters”: words with 2 letters TT and 2 letters AA: ((4!)/(2!2!))=6 words with 2 letters TT, other letters are different: C_2 ^6 ×((4!)/(2!))=180 words with 2 letters AA, other letters are different: C_2 ^6 ×((4!)/(2!))=180 words with different letters: 7×6×5×4=840 ⇒totally 6+180×2+840=1206 ways

$${case}\:``{words}\:{with}\:\mathrm{4}\:{letters}'': \\ $$$${words}\:{with}\:\mathrm{2}\:{letters}\:{TT}\:{and}\:\mathrm{2}\:{letters}\:{AA}: \\ $$$$\frac{\mathrm{4}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{6} \\ $$$$ \\ $$$${words}\:{with}\:\mathrm{2}\:{letters}\:{TT},\:{other}\:{letters}\:{are}\:{different}: \\ $$$${C}_{\mathrm{2}} ^{\mathrm{6}} ×\frac{\mathrm{4}!}{\mathrm{2}!}=\mathrm{180} \\ $$$$ \\ $$$${words}\:{with}\:\mathrm{2}\:{letters}\:{AA},\:{other}\:{letters}\:{are}\:{different}: \\ $$$${C}_{\mathrm{2}} ^{\mathrm{6}} ×\frac{\mathrm{4}!}{\mathrm{2}!}=\mathrm{180} \\ $$$$ \\ $$$${words}\:{with}\:{different}\:{letters}: \\ $$$$\mathrm{7}×\mathrm{6}×\mathrm{5}×\mathrm{4}=\mathrm{840} \\ $$$$ \\ $$$$\Rightarrow{totally}\:\mathrm{6}+\mathrm{180}×\mathrm{2}+\mathrm{840}=\mathrm{1206}\:{ways} \\ $$

Commented by afachri last updated on 10/Feb/19

aahh i did it wrong. thank to illuminate me Mr W.

$$\boldsymbol{\mathrm{aahh}}\:\boldsymbol{\mathrm{i}}\:\boldsymbol{\mathrm{did}}\:\boldsymbol{\mathrm{it}}\:\boldsymbol{\mathrm{wrong}}.\:\boldsymbol{\mathrm{thank}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{illuminate}} \\ $$$$\boldsymbol{\mathrm{me}}\:\boldsymbol{\mathrm{Mr}}\:\boldsymbol{\mathrm{W}}. \\ $$

Commented by afachri last updated on 10/Feb/19

so Sir, why the 5 letters case is not counted ??

$$\boldsymbol{\mathrm{so}}\:\boldsymbol{{S}}{ir},\:{why}\:{the}\:\mathrm{5}\:{letters}\:{case}\:{is}\:{not} \\ $$$${counted}\:?? \\ $$

Commented by mr W last updated on 10/Feb/19

i have not checked the other cases.

$${i}\:{have}\:{not}\:{checked}\:{the}\:{other}\:{cases}. \\ $$

Commented by afachri last updated on 11/Feb/19

yes Sir

$${yes}\:{Sir} \\ $$

Answered by mr W last updated on 12/Feb/19

we have totally 9 letters: TT AA INKUR − words with 2 letters with same letters: 2 ways, i.e. TT and AA with different letters: P_2 ^7 =7×6=42 ways ⇒2+42=44 ways − words with 3 letters with 2 same letters TT: C_1 ^6 ×((3!)/(2!))=18 ways with 2 same letters AA: C_1 ^6 ×((3!)/(2!))=18 ways with different letters: P_3 ^7 =7×6×5=210 ways ⇒2×18+210=246 ways − words with 4 letters with 2 TT and 2 AA: ((4!)/(2!2!))=6 ways with 2 TT and 2 other different letters: C_2 ^6 ×((4!)/(2!))=180 ways with 2 AA and 2 other different letters: C_2 ^6 ×((4!)/(2!))=180 ways with 4 different letters: P_4 ^7 =7×6×5×4=840 ways ⇒6+2×180+840=1206 ways − words with 5 letters TT+AA+x: C_1 ^5 ×((5!)/(2!2!))=150 ways TT+xxx: C_3 ^6 ×((5!)/(2!))=1200 ways AA+xxx: C_3 ^6 ×((5!)/(2!))=1200 ways with 5 different letters: P_5 ^7 =2520 ways ⇒150+2×1200+2520=5070 ways − words with 6 letters TT+AA+xx: C_2 ^5 ×((6!)/(2!2!))=1800 ways TT+xxxx: C_4 ^6 ×((6!)/(2!))=5400 ways AA+xxxx: C_4 ^6 ×((6!)/(2!))=5400 ways with 6 different letters: P_6 ^7 =5040 ways ⇒1800+2×5400+5040=17640 ways − words with 7 letters TT+AA+xxx: C_3 ^5 ×((7!)/(2!2!))=12600 ways TT+xxxxx: C_5 ^6 ×((7!)/(2!))=15120 ways AA+xxxxx: C_5 ^6 ×((7!)/(2!))=15120 ways with 7 different letters: P_7 ^7 =5040 ways ⇒12600+2×15120+5040=47880 ways − words with 8 letters TT+AA+xxxx: C_4 ^5 ×((8!)/(2!2!))=50400 ways TT+xxxxxx: C_6 ^6 ×((8!)/(2!))=20160 ways AA+xxxxxx: C_6 ^6 ×((8!)/(2!))=20160 ways ⇒50400+2×20160=90720 − words with 9 letters ⇒((9!)/(2!2!))=90720 sum: 44+246+1206+5070+17640+47880+90720+90720 =253526 ways

$${we}\:{have}\:{totally}\:\mathrm{9}\:{letters}: \\ $$$${TT} \\ $$$${AA} \\ $$$${INKUR} \\ $$$$ \\ $$$$−\:{words}\:{with}\:\mathrm{2}\:{letters} \\ $$$${with}\:{same}\:{letters}:\:\mathrm{2}\:{ways},\:{i}.{e}.\:{TT}\:{and}\:{AA} \\ $$$${with}\:{different}\:{letters}:\:{P}_{\mathrm{2}} ^{\mathrm{7}} =\mathrm{7}×\mathrm{6}=\mathrm{42}\:{ways}\: \\ $$$$\Rightarrow\mathrm{2}+\mathrm{42}=\mathrm{44}\:{ways} \\ $$$$ \\ $$$$−\:{words}\:{with}\:\mathrm{3}\:{letters} \\ $$$${with}\:\mathrm{2}\:{same}\:{letters}\:{TT}:\:{C}_{\mathrm{1}} ^{\mathrm{6}} ×\frac{\mathrm{3}!}{\mathrm{2}!}=\mathrm{18}\:{ways} \\ $$$${with}\:\mathrm{2}\:{same}\:{letters}\:{AA}:\:{C}_{\mathrm{1}} ^{\mathrm{6}} ×\frac{\mathrm{3}!}{\mathrm{2}!}=\mathrm{18}\:{ways} \\ $$$${with}\:{different}\:{letters}:\:{P}_{\mathrm{3}} ^{\mathrm{7}} =\mathrm{7}×\mathrm{6}×\mathrm{5}=\mathrm{210}\:{ways} \\ $$$$\Rightarrow\mathrm{2}×\mathrm{18}+\mathrm{210}=\mathrm{246}\:{ways} \\ $$$$ \\ $$$$−\:{words}\:{with}\:\mathrm{4}\:{letters} \\ $$$${with}\:\mathrm{2}\:{TT}\:{and}\:\mathrm{2}\:{AA}:\:\frac{\mathrm{4}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{6}\:{ways} \\ $$$${with}\:\mathrm{2}\:{TT}\:{and}\:\mathrm{2}\:{other}\:{different}\:{letters}:\:{C}_{\mathrm{2}} ^{\mathrm{6}} ×\frac{\mathrm{4}!}{\mathrm{2}!}=\mathrm{180}\:{ways} \\ $$$${with}\:\mathrm{2}\:{AA}\:{and}\:\mathrm{2}\:{other}\:{different}\:{letters}:\:{C}_{\mathrm{2}} ^{\mathrm{6}} ×\frac{\mathrm{4}!}{\mathrm{2}!}=\mathrm{180}\:{ways} \\ $$$${with}\:\mathrm{4}\:{different}\:{letters}:\:{P}_{\mathrm{4}} ^{\mathrm{7}} =\mathrm{7}×\mathrm{6}×\mathrm{5}×\mathrm{4}=\mathrm{840}\:{ways} \\ $$$$\Rightarrow\mathrm{6}+\mathrm{2}×\mathrm{180}+\mathrm{840}=\mathrm{1206}\:{ways} \\ $$$$ \\ $$$$−\:{words}\:{with}\:\mathrm{5}\:{letters} \\ $$$${TT}+{AA}+{x}:\:{C}_{\mathrm{1}} ^{\mathrm{5}} ×\frac{\mathrm{5}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{150}\:{ways} \\ $$$${TT}+{xxx}:\:{C}_{\mathrm{3}} ^{\mathrm{6}} ×\frac{\mathrm{5}!}{\mathrm{2}!}=\mathrm{1200}\:{ways} \\ $$$${AA}+{xxx}:\:{C}_{\mathrm{3}} ^{\mathrm{6}} ×\frac{\mathrm{5}!}{\mathrm{2}!}=\mathrm{1200}\:{ways} \\ $$$${with}\:\mathrm{5}\:{different}\:{letters}:\:{P}_{\mathrm{5}} ^{\mathrm{7}} =\mathrm{2520}\:{ways} \\ $$$$\Rightarrow\mathrm{150}+\mathrm{2}×\mathrm{1200}+\mathrm{2520}=\mathrm{5070}\:{ways} \\ $$$$ \\ $$$$−\:{words}\:{with}\:\mathrm{6}\:{letters} \\ $$$${TT}+{AA}+{xx}:\:{C}_{\mathrm{2}} ^{\mathrm{5}} ×\frac{\mathrm{6}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{1800}\:{ways} \\ $$$${TT}+{xxxx}:\:{C}_{\mathrm{4}} ^{\mathrm{6}} ×\frac{\mathrm{6}!}{\mathrm{2}!}=\mathrm{5400}\:{ways} \\ $$$${AA}+{xxxx}:\:{C}_{\mathrm{4}} ^{\mathrm{6}} ×\frac{\mathrm{6}!}{\mathrm{2}!}=\mathrm{5400}\:{ways} \\ $$$${with}\:\mathrm{6}\:{different}\:{letters}:\:{P}_{\mathrm{6}} ^{\mathrm{7}} =\mathrm{5040}\:{ways} \\ $$$$\Rightarrow\mathrm{1800}+\mathrm{2}×\mathrm{5400}+\mathrm{5040}=\mathrm{17640}\:{ways} \\ $$$$ \\ $$$$−\:{words}\:{with}\:\mathrm{7}\:{letters} \\ $$$${TT}+{AA}+{xxx}:\:{C}_{\mathrm{3}} ^{\mathrm{5}} ×\frac{\mathrm{7}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{12600}\:{ways} \\ $$$${TT}+{xxxxx}:\:{C}_{\mathrm{5}} ^{\mathrm{6}} ×\frac{\mathrm{7}!}{\mathrm{2}!}=\mathrm{15120}\:{ways} \\ $$$${AA}+{xxxxx}:\:{C}_{\mathrm{5}} ^{\mathrm{6}} ×\frac{\mathrm{7}!}{\mathrm{2}!}=\mathrm{15120}\:{ways} \\ $$$${with}\:\mathrm{7}\:{different}\:{letters}:\:{P}_{\mathrm{7}} ^{\mathrm{7}} =\mathrm{5040}\:{ways} \\ $$$$\Rightarrow\mathrm{12600}+\mathrm{2}×\mathrm{15120}+\mathrm{5040}=\mathrm{47880}\:{ways} \\ $$$$ \\ $$$$−\:{words}\:{with}\:\mathrm{8}\:{letters} \\ $$$${TT}+{AA}+{xxxx}:\:{C}_{\mathrm{4}} ^{\mathrm{5}} ×\frac{\mathrm{8}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{50400}\:{ways} \\ $$$${TT}+{xxxxxx}:\:{C}_{\mathrm{6}} ^{\mathrm{6}} ×\frac{\mathrm{8}!}{\mathrm{2}!}=\mathrm{20160}\:{ways} \\ $$$${AA}+{xxxxxx}:\:{C}_{\mathrm{6}} ^{\mathrm{6}} ×\frac{\mathrm{8}!}{\mathrm{2}!}=\mathrm{20160}\:{ways} \\ $$$$\Rightarrow\mathrm{50400}+\mathrm{2}×\mathrm{20160}=\mathrm{90720} \\ $$$$ \\ $$$$−\:{words}\:{with}\:\mathrm{9}\:{letters} \\ $$$$\Rightarrow\frac{\mathrm{9}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{90720} \\ $$$$ \\ $$$${sum}:\:\mathrm{44}+\mathrm{246}+\mathrm{1206}+\mathrm{5070}+\mathrm{17640}+\mathrm{47880}+\mathrm{90720}+\mathrm{90720} \\ $$$$=\mathrm{253526}\:{ways} \\ $$

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