document.write( "Question 417988: Using the letters MAMMAL find:
\n" ); document.write( "a) how many different ways to arrange the letters
\n" ); document.write( "b) If we insist that the letter M come first and last, how many ways are there? \n" ); document.write( "
Algebra.Com's Answer #292629 by Edwin McCravy(20063)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "If we could distinguish all 6 letters we would have 7! or 5040.\r\n" );
document.write( "If we could not distinguish the 3 M's but could distinguish the 2 A's,\r\n" );
document.write( "then since there are 3 positions in which the M's could be placed, this \r\n" );
document.write( "is 3! times too many, So we would divide that by 3! and get 840 ways, if \r\n" );
document.write( "the M's are not distinguishable but the A's are.\r\n" );
document.write( "But since the A's are also indistinguishable, we must also divide that\r\n" );
document.write( "by 2!.  So there are\r\n" );
document.write( "\r\n" );
document.write( "6!/(3!2!) = 720/(6*2) = 720/12 = 60 ways. Here they are: \r\n" );
document.write( "  \r\n" );
document.write( " 1. MAMMAL\r\n" );
document.write( " 2. MAMMLA\r\n" );
document.write( " 3. MAMAML\r\n" );
document.write( " 4. MAMALM\r\n" );
document.write( " 5. MAMLMA\r\n" );
document.write( " 6. MAMLAM\r\n" );
document.write( " 7. MAAMML\r\n" );
document.write( " 8. MAAMLM\r\n" );
document.write( " 9. MAALMM\r\n" );
document.write( "10. MALMMA\r\n" );
document.write( "11. MALMAM\r\n" );
document.write( "12. MALAMM\r\n" );
document.write( "13. MMAMAL\r\n" );
document.write( "14. MMAMLA\r\n" );
document.write( "15. MMAAML\r\n" );
document.write( "16. MMAALM\r\n" );
document.write( "17. MMALMA\r\n" );
document.write( "18. MMALAM\r\n" );
document.write( "19. MMMAAL\r\n" );
document.write( "20. MMMALA\r\n" );
document.write( "21. MMMLAA\r\n" );
document.write( "22. MMLAMA\r\n" );
document.write( "23. MMLAAM\r\n" );
document.write( "24. MMLMAA\r\n" );
document.write( "25. MLAMMA\r\n" );
document.write( "26. MLAMAM\r\n" );
document.write( "27. MLAAMM\r\n" );
document.write( "28. MLMAMA\r\n" );
document.write( "29. MLMAAM\r\n" );
document.write( "30. MLMMAA\r\n" );
document.write( "31. AMMMAL\r\n" );
document.write( "32. AMMMLA\r\n" );
document.write( "33. AMMAML\r\n" );
document.write( "34. AMMALM\r\n" );
document.write( "35. AMMLMA\r\n" );
document.write( "36. AMMLAM\r\n" );
document.write( "37. AMAMML\r\n" );
document.write( "38. AMAMLM\r\n" );
document.write( "39. AMALMM\r\n" );
document.write( "40. AMLMMA\r\n" );
document.write( "41. AMLMAM\r\n" );
document.write( "42. AMLAMM\r\n" );
document.write( "43. AAMMML\r\n" );
document.write( "44. AAMMLM\r\n" );
document.write( "45. AAMLMM\r\n" );
document.write( "46. AALMMM\r\n" );
document.write( "47. ALMMMA\r\n" );
document.write( "48. ALMMAM\r\n" );
document.write( "49. ALMAMM\r\n" );
document.write( "50. ALAMMM\r\n" );
document.write( "51. LMAMMA\r\n" );
document.write( "52. LMAMAM\r\n" );
document.write( "53. LMAAMM\r\n" );
document.write( "54. LMMAMA\r\n" );
document.write( "55. LMMAAM\r\n" );
document.write( "56. LMMMAA\r\n" );
document.write( "57. LAMMMA\r\n" );
document.write( "58. LAMMAM\r\n" );
document.write( "59. LAMAMM\r\n" );
document.write( "60. LAAMMM\r\n" );
document.write( "\r\n" );
document.write( "If we require that M's come first and last:\r\n" );
document.write( "\r\n" );
document.write( "M _ _ _ _ M\r\n" );
document.write( "\r\n" );
document.write( "Then we just have to fill in the middle 4 blanks with A,M,A,L,\r\n" );
document.write( "using the same reasoning as above:\r\n" );
document.write( "\r\n" );
document.write( "So the answer is 4!/2! = 24/2 = 12 ways. They are indicated in\r\n" );
document.write( "red in the above list.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );