Repeated alphabet in n^r












0












$begingroup$


Q: How many ways to form a word of 3 letters with English alphabet (repetition allowed)?



A: 26^3



but shouldn't we account for the repeated alphabet? like aab, and aab is the same?



Why isn't the question solved in the same way like "How many ways to arrange the word POOP"



which the answer is 4!/(2!2!)










share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    Q: How many ways to form a word of 3 letters with English alphabet (repetition allowed)?



    A: 26^3



    but shouldn't we account for the repeated alphabet? like aab, and aab is the same?



    Why isn't the question solved in the same way like "How many ways to arrange the word POOP"



    which the answer is 4!/(2!2!)










    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      Q: How many ways to form a word of 3 letters with English alphabet (repetition allowed)?



      A: 26^3



      but shouldn't we account for the repeated alphabet? like aab, and aab is the same?



      Why isn't the question solved in the same way like "How many ways to arrange the word POOP"



      which the answer is 4!/(2!2!)










      share|cite|improve this question











      $endgroup$




      Q: How many ways to form a word of 3 letters with English alphabet (repetition allowed)?



      A: 26^3



      but shouldn't we account for the repeated alphabet? like aab, and aab is the same?



      Why isn't the question solved in the same way like "How many ways to arrange the word POOP"



      which the answer is 4!/(2!2!)







      permutations






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 22 '18 at 7:35







      thikin hax

















      asked Dec 22 '18 at 7:28









      thikin haxthikin hax

      11




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          $begingroup$

          We already are accounting for the repeated alphabet. That $26^3$ comes from answering three questions: What is the first letter? What is the second letter? What is the third letter?

          Those three questions are allowed to have the same answers, as in your example "aab". That's $26$ possibilities for each question, and each answer is fully independent of the others.



          Arranging the letters of a fixed word "deed"? We can ask which letter goes in each place, but those choices are tied together by the need to use each of that original word's letters. The numerator $4cdot 3cdot 2cdot 1$ comes from our choices being reduced at every step to whatever we haven't used yet, and the denominator $2times 2$ comes from letters of the same type being indistinguishable; $d_1e_1e_2d_2$ and $d_2e_1e_2d_1$ are the same words.






          share|cite|improve this answer









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            $begingroup$

            We already are accounting for the repeated alphabet. That $26^3$ comes from answering three questions: What is the first letter? What is the second letter? What is the third letter?

            Those three questions are allowed to have the same answers, as in your example "aab". That's $26$ possibilities for each question, and each answer is fully independent of the others.



            Arranging the letters of a fixed word "deed"? We can ask which letter goes in each place, but those choices are tied together by the need to use each of that original word's letters. The numerator $4cdot 3cdot 2cdot 1$ comes from our choices being reduced at every step to whatever we haven't used yet, and the denominator $2times 2$ comes from letters of the same type being indistinguishable; $d_1e_1e_2d_2$ and $d_2e_1e_2d_1$ are the same words.






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              We already are accounting for the repeated alphabet. That $26^3$ comes from answering three questions: What is the first letter? What is the second letter? What is the third letter?

              Those three questions are allowed to have the same answers, as in your example "aab". That's $26$ possibilities for each question, and each answer is fully independent of the others.



              Arranging the letters of a fixed word "deed"? We can ask which letter goes in each place, but those choices are tied together by the need to use each of that original word's letters. The numerator $4cdot 3cdot 2cdot 1$ comes from our choices being reduced at every step to whatever we haven't used yet, and the denominator $2times 2$ comes from letters of the same type being indistinguishable; $d_1e_1e_2d_2$ and $d_2e_1e_2d_1$ are the same words.






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                We already are accounting for the repeated alphabet. That $26^3$ comes from answering three questions: What is the first letter? What is the second letter? What is the third letter?

                Those three questions are allowed to have the same answers, as in your example "aab". That's $26$ possibilities for each question, and each answer is fully independent of the others.



                Arranging the letters of a fixed word "deed"? We can ask which letter goes in each place, but those choices are tied together by the need to use each of that original word's letters. The numerator $4cdot 3cdot 2cdot 1$ comes from our choices being reduced at every step to whatever we haven't used yet, and the denominator $2times 2$ comes from letters of the same type being indistinguishable; $d_1e_1e_2d_2$ and $d_2e_1e_2d_1$ are the same words.






                share|cite|improve this answer









                $endgroup$



                We already are accounting for the repeated alphabet. That $26^3$ comes from answering three questions: What is the first letter? What is the second letter? What is the third letter?

                Those three questions are allowed to have the same answers, as in your example "aab". That's $26$ possibilities for each question, and each answer is fully independent of the others.



                Arranging the letters of a fixed word "deed"? We can ask which letter goes in each place, but those choices are tied together by the need to use each of that original word's letters. The numerator $4cdot 3cdot 2cdot 1$ comes from our choices being reduced at every step to whatever we haven't used yet, and the denominator $2times 2$ comes from letters of the same type being indistinguishable; $d_1e_1e_2d_2$ and $d_2e_1e_2d_1$ are the same words.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 22 '18 at 7:39









                jmerryjmerry

                12.6k1628




                12.6k1628






























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