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

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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?
Howmanywordswithatleast2letterscanbeformedusingthelettersfromTINKUTARA?
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.
thankyousir!pleaserecheckyourresults.forexamplewordswith2letters:with2sameletters:2ways,i.e.TTandAA.with2differentletters:wecanchoose2lettersfrom7letters(i.e.TINKUAR),andwehave7×6=42ways.thatistosaythereare2+42=44waystoformwordswith2letters,not18ways.
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“wordswith3letters″:wordswith2lettersTT:TTx,TxT,xTT⇒3×6=18wayswordswith2lettersAA:AAx,AxA,xAA⇒3×6=18wayswordswith3differentletters:7×6×5=210ways⇒totally2×18+210=246ways,not126ways.
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“wordswith4letters″:wordswith2lettersTTand2lettersAA:4!2!2!=6wordswith2lettersTT,otherlettersaredifferent:C26×4!2!=180wordswith2lettersAA,otherlettersaredifferent:C26×4!2!=180wordswithdifferentletters:7×6×5×4=840⇒totally6+180×2+840=1206ways
Commented by afachri last updated on 10/Feb/19
aahh i did it wrong. thank to illuminate me Mr W.
aahhididitwrong.thanktoilluminatemeMrW.
Commented by afachri last updated on 10/Feb/19
so Sir, why the 5 letters case is not counted ??
soSir,whythe5letterscaseisnotcounted??
Commented by mr W last updated on 10/Feb/19
i have not checked the other cases.
ihavenotcheckedtheothercases.
Commented by afachri last updated on 11/Feb/19
yes Sir
yesSir
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
wehavetotally9letters:TTAAINKUR−wordswith2letterswithsameletters:2ways,i.e.TTandAAwithdifferentletters:P27=7×6=42ways⇒2+42=44ways−wordswith3letterswith2samelettersTT:C16×3!2!=18wayswith2samelettersAA:C16×3!2!=18wayswithdifferentletters:P37=7×6×5=210ways⇒2×18+210=246ways−wordswith4letterswith2TTand2AA:4!2!2!=6wayswith2TTand2otherdifferentletters:C26×4!2!=180wayswith2AAand2otherdifferentletters:C26×4!2!=180wayswith4differentletters:P47=7×6×5×4=840ways⇒6+2×180+840=1206ways−wordswith5lettersTT+AA+x:C15×5!2!2!=150waysTT+xxx:C36×5!2!=1200waysAA+xxx:C36×5!2!=1200wayswith5differentletters:P57=2520ways⇒150+2×1200+2520=5070ways−wordswith6lettersTT+AA+xx:C25×6!2!2!=1800waysTT+xxxx:C46×6!2!=5400waysAA+xxxx:C46×6!2!=5400wayswith6differentletters:P67=5040ways⇒1800+2×5400+5040=17640ways−wordswith7lettersTT+AA+xxx:C35×7!2!2!=12600waysTT+xxxxx:C56×7!2!=15120waysAA+xxxxx:C56×7!2!=15120wayswith7differentletters:P77=5040ways⇒12600+2×15120+5040=47880ways−wordswith8lettersTT+AA+xxxx:C45×8!2!2!=50400waysTT+xxxxxx:C66×8!2!=20160waysAA+xxxxxx:C66×8!2!=20160ways⇒50400+2×20160=90720−wordswith9letters⇒9!2!2!=90720sum:44+246+1206+5070+17640+47880+90720+90720=253526ways

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