How many 6-letter arrangements can be made from the letters in STAMPEDE?











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I tried casework for this. For the 2 Es case, I chose 2 non-Es to exclude then arranged the remaining 6 letters. Please pardon the formatting; I’m on my phone.



n(0 Es)+n(1 E)+n(2 Es)
=6! + 7P6 + (6C2)(6!/2!)



Can someone let me know if my work is right? Thank you!










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    up vote
    1
    down vote

    favorite












    I tried casework for this. For the 2 Es case, I chose 2 non-Es to exclude then arranged the remaining 6 letters. Please pardon the formatting; I’m on my phone.



    n(0 Es)+n(1 E)+n(2 Es)
    =6! + 7P6 + (6C2)(6!/2!)



    Can someone let me know if my work is right? Thank you!










    share|cite|improve this question


























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I tried casework for this. For the 2 Es case, I chose 2 non-Es to exclude then arranged the remaining 6 letters. Please pardon the formatting; I’m on my phone.



      n(0 Es)+n(1 E)+n(2 Es)
      =6! + 7P6 + (6C2)(6!/2!)



      Can someone let me know if my work is right? Thank you!










      share|cite|improve this question















      I tried casework for this. For the 2 Es case, I chose 2 non-Es to exclude then arranged the remaining 6 letters. Please pardon the formatting; I’m on my phone.



      n(0 Es)+n(1 E)+n(2 Es)
      =6! + 7P6 + (6C2)(6!/2!)



      Can someone let me know if my work is right? Thank you!







      combinatorics permutations






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      edited Nov 18 at 0:23









      N. F. Taussig

      42.8k93254




      42.8k93254










      asked Nov 17 at 22:01









      Daniel Nguyen

      734




      734






















          1 Answer
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          accepted










          Almost.



          Number of arrangements of six letters with no Es: There are six letters other than E in STAMPEDE. Since the letters are distinct, they can be arranged in $6!$ ways.



          Number of arrangements of six letters with exactly one E: We must choose which five of the other six letters will be used with the E. Since the six letters are all distinct, they can be arranged in $6!$ ways. Thus, the number of arrangements with exactly one E is
          $$binom{6}{5}6!$$
          This where you made your mistake.



          Number of arrangements of six letters with two Es: We must choose which four of the other six letters will be used with the Es. Thus, we must arrange six letters, two of which are Es and with each of the others appearing exactly once. We choose two of the six positions for the Es. The remaining four distinct letters can be arranged in the remaining four positions in $4!$ ways. Hence, there are
          $$binom{6}{4}binom{6}{2}4! = frac{6!}{4!2!} cdot frac{6!}{2!4!} cdot 4! = binom{6}{2} cdot frac{6!}{2!}$$
          as you found.



          Total: $$6! + binom{6}{5}6! + binom{6}{4}binom{6}{2}4!$$






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          • 1




            Makes sense! Thanks for the correction on the second term!
            – Daniel Nguyen
            Nov 18 at 8:02











          Your Answer





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          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          2
          down vote



          accepted










          Almost.



          Number of arrangements of six letters with no Es: There are six letters other than E in STAMPEDE. Since the letters are distinct, they can be arranged in $6!$ ways.



          Number of arrangements of six letters with exactly one E: We must choose which five of the other six letters will be used with the E. Since the six letters are all distinct, they can be arranged in $6!$ ways. Thus, the number of arrangements with exactly one E is
          $$binom{6}{5}6!$$
          This where you made your mistake.



          Number of arrangements of six letters with two Es: We must choose which four of the other six letters will be used with the Es. Thus, we must arrange six letters, two of which are Es and with each of the others appearing exactly once. We choose two of the six positions for the Es. The remaining four distinct letters can be arranged in the remaining four positions in $4!$ ways. Hence, there are
          $$binom{6}{4}binom{6}{2}4! = frac{6!}{4!2!} cdot frac{6!}{2!4!} cdot 4! = binom{6}{2} cdot frac{6!}{2!}$$
          as you found.



          Total: $$6! + binom{6}{5}6! + binom{6}{4}binom{6}{2}4!$$






          share|cite|improve this answer

















          • 1




            Makes sense! Thanks for the correction on the second term!
            – Daniel Nguyen
            Nov 18 at 8:02















          up vote
          2
          down vote



          accepted










          Almost.



          Number of arrangements of six letters with no Es: There are six letters other than E in STAMPEDE. Since the letters are distinct, they can be arranged in $6!$ ways.



          Number of arrangements of six letters with exactly one E: We must choose which five of the other six letters will be used with the E. Since the six letters are all distinct, they can be arranged in $6!$ ways. Thus, the number of arrangements with exactly one E is
          $$binom{6}{5}6!$$
          This where you made your mistake.



          Number of arrangements of six letters with two Es: We must choose which four of the other six letters will be used with the Es. Thus, we must arrange six letters, two of which are Es and with each of the others appearing exactly once. We choose two of the six positions for the Es. The remaining four distinct letters can be arranged in the remaining four positions in $4!$ ways. Hence, there are
          $$binom{6}{4}binom{6}{2}4! = frac{6!}{4!2!} cdot frac{6!}{2!4!} cdot 4! = binom{6}{2} cdot frac{6!}{2!}$$
          as you found.



          Total: $$6! + binom{6}{5}6! + binom{6}{4}binom{6}{2}4!$$






          share|cite|improve this answer

















          • 1




            Makes sense! Thanks for the correction on the second term!
            – Daniel Nguyen
            Nov 18 at 8:02













          up vote
          2
          down vote



          accepted







          up vote
          2
          down vote



          accepted






          Almost.



          Number of arrangements of six letters with no Es: There are six letters other than E in STAMPEDE. Since the letters are distinct, they can be arranged in $6!$ ways.



          Number of arrangements of six letters with exactly one E: We must choose which five of the other six letters will be used with the E. Since the six letters are all distinct, they can be arranged in $6!$ ways. Thus, the number of arrangements with exactly one E is
          $$binom{6}{5}6!$$
          This where you made your mistake.



          Number of arrangements of six letters with two Es: We must choose which four of the other six letters will be used with the Es. Thus, we must arrange six letters, two of which are Es and with each of the others appearing exactly once. We choose two of the six positions for the Es. The remaining four distinct letters can be arranged in the remaining four positions in $4!$ ways. Hence, there are
          $$binom{6}{4}binom{6}{2}4! = frac{6!}{4!2!} cdot frac{6!}{2!4!} cdot 4! = binom{6}{2} cdot frac{6!}{2!}$$
          as you found.



          Total: $$6! + binom{6}{5}6! + binom{6}{4}binom{6}{2}4!$$






          share|cite|improve this answer












          Almost.



          Number of arrangements of six letters with no Es: There are six letters other than E in STAMPEDE. Since the letters are distinct, they can be arranged in $6!$ ways.



          Number of arrangements of six letters with exactly one E: We must choose which five of the other six letters will be used with the E. Since the six letters are all distinct, they can be arranged in $6!$ ways. Thus, the number of arrangements with exactly one E is
          $$binom{6}{5}6!$$
          This where you made your mistake.



          Number of arrangements of six letters with two Es: We must choose which four of the other six letters will be used with the Es. Thus, we must arrange six letters, two of which are Es and with each of the others appearing exactly once. We choose two of the six positions for the Es. The remaining four distinct letters can be arranged in the remaining four positions in $4!$ ways. Hence, there are
          $$binom{6}{4}binom{6}{2}4! = frac{6!}{4!2!} cdot frac{6!}{2!4!} cdot 4! = binom{6}{2} cdot frac{6!}{2!}$$
          as you found.



          Total: $$6! + binom{6}{5}6! + binom{6}{4}binom{6}{2}4!$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 18 at 0:21









          N. F. Taussig

          42.8k93254




          42.8k93254








          • 1




            Makes sense! Thanks for the correction on the second term!
            – Daniel Nguyen
            Nov 18 at 8:02














          • 1




            Makes sense! Thanks for the correction on the second term!
            – Daniel Nguyen
            Nov 18 at 8:02








          1




          1




          Makes sense! Thanks for the correction on the second term!
          – Daniel Nguyen
          Nov 18 at 8:02




          Makes sense! Thanks for the correction on the second term!
          – Daniel Nguyen
          Nov 18 at 8:02


















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