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

$$Howmanywordswithatleast2letters$$$$canbeformedusingthelettersfrom$$$$TINKUTARA?$$

Answered by afachri last updated on 09/Feb/19

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

$$thankyousir!$$$$pleaserecheckyourresults.$$$$forexamplewordswith2letters:$$$$$$$$with2sameletters:2ways,i.e.TTandAA.$$$$$$$$with2differentletters:wecanchoose$$$$2lettersfrom7letters(i.e.TINKUAR),$$$$andwehave7\times 6=42ways.$$$$$$$$thatistosaythereare2+42=44ways$$$$toformwordswith2letters,not18ways.$$

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

$$case‘‘wordswith3{letters}^{″}:$$$$wordswith2lettersTT:TTx,TxT,xTT\Rightarrow 3\times 6=18ways$$$$wordswith2lettersAA:AAx,AxA,xAA\Rightarrow 3\times 6=18ways$$$$wordswith3differentletters:7\times 6\times 5=210ways$$$$\Rightarrow totally2\times 18+210=246ways,not126ways.$$

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

$$case‘‘wordswith4{letters}^{″}:$$$$wordswith2lettersTTand2lettersAA:$$$$\frac{4!}{2!2!}=6$$$$$$$$wordswith2lettersTT,otherlettersaredifferent:$$$$C_2^6\times \frac{4!}{2!}=180$$$$$$$$wordswith2lettersAA,otherlettersaredifferent:$$$$C_2^6\times \frac{4!}{2!}=180$$$$$$$$wordswithdifferentletters:$$$$7\times 6\times 5\times 4=840$$$$$$$$\Rightarrow totally6+180\times 2+840=1206ways$$

Commented by afachri last updated on 10/Feb/19

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

Commented by afachri last updated on 10/Feb/19

$$\mathbf{so}\mathit{S}ir,whythe5letterscaseisnot$$$$counted??$$

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

$$ihavenotcheckedtheothercases.$$

Commented by afachri last updated on 11/Feb/19

$$yesSir$$

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

$$wehavetotally9letters:$$$$TT$$$$AA$$$$INKUR$$$$$$$$-wordswith2letters$$$$withsameletters:2ways,i.e.TTandAA$$$$withdifferentletters:P_2^7=7\times 6=42ways$$$$\Rightarrow 2+42=44ways$$$$$$$$-wordswith3letters$$$$with2samelettersTT:C_1^6\times \frac{3!}{2!}=18ways$$$$with2samelettersAA:C_1^6\times \frac{3!}{2!}=18ways$$$$withdifferentletters:P_3^7=7\times 6\times 5=210ways$$$$\Rightarrow 2\times 18+210=246ways$$$$$$$$-wordswith4letters$$$$with2TTand2AA:\frac{4!}{2!2!}=6ways$$$$with2TTand2otherdifferentletters:C_2^6\times \frac{4!}{2!}=180ways$$$$with2AAand2otherdifferentletters:C_2^6\times \frac{4!}{2!}=180ways$$$$with4differentletters:P_4^7=7\times 6\times 5\times 4=840ways$$$$\Rightarrow 6+2\times 180+840=1206ways$$$$$$$$-wordswith5letters$$$$TT+AA+x:C_1^5\times \frac{5!}{2!2!}=150ways$$$$TT+xxx:C_3^6\times \frac{5!}{2!}=1200ways$$$$AA+xxx:C_3^6\times \frac{5!}{2!}=1200ways$$$$with5differentletters:P_5^7=2520ways$$$$\Rightarrow 150+2\times 1200+2520=5070ways$$$$$$$$-wordswith6letters$$$$TT+AA+xx:C_2^5\times \frac{6!}{2!2!}=1800ways$$$$TT+xxxx:C_4^6\times \frac{6!}{2!}=5400ways$$$$AA+xxxx:C_4^6\times \frac{6!}{2!}=5400ways$$$$with6differentletters:P_6^7=5040ways$$$$\Rightarrow 1800+2\times 5400+5040=17640ways$$$$$$$$-wordswith7letters$$$$TT+AA+xxx:C_3^5\times \frac{7!}{2!2!}=12600ways$$$$TT+xxxxx:C_5^6\times \frac{7!}{2!}=15120ways$$$$AA+xxxxx:C_5^6\times \frac{7!}{2!}=15120ways$$$$with7differentletters:P_7^7=5040ways$$$$\Rightarrow 12600+2\times 15120+5040=47880ways$$$$$$$$-wordswith8letters$$$$TT+AA+xxxx:C_4^5\times \frac{8!}{2!2!}=50400ways$$$$TT+xxxxxx:C_6^6\times \frac{8!}{2!}=20160ways$$$$AA+xxxxxx:C_6^6\times \frac{8!}{2!}=20160ways$$$$\Rightarrow 50400+2\times 20160=90720$$$$$$$$-wordswith9letters$$$$\Rightarrow \frac{9!}{2!2!}=90720$$$$$$$$sum:44+246+1206+5070+17640+47880+90720+90720$$$$=253526ways$$

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