Date: Wed, 14 Nov 2007 22:18:51 -0600 (CST) From: Mark Jacobson Subject: Re: Questions and answers about the 2nd MIDTERM exam... > Quick question for you. > > 9. How many 3 letter words can be made from the letters in PANTHERS? Words > like SSS would be okay, because repetition of a letter is allowed. > The correct answer is P(8,3) the way this question is worded. However, your question brings up another great question, IF WE ASSUME YOU CAN REPEAT USE OF A LETTER. This would be like ASKING HOW MANY WAYS CAN WE HAVE our class of 30 students have class officers: PRESIDENT Vice President Treasurer Secretary It is P(30, 4) ways or 30*29*28*27 ways. If we say, however, anyone can hold more than one office, oh, lets say anyone in the class can hold as many offices as they want, then the answer is: 4 30 = 30 * 30 * 30 * 30 ways > Would this be 8*8*8=512 because I can reuse each of the 8 choices for each of > the three letters? Yes, choose P as the first letter and you still have 8 choices left for the second letter, if repetitions are allowed, and 8 choices left for the 3rd letter, IF REPEATED letters are allowed. ----------- How many ways can you arrange any 3 letters from PANTHERS is P(8, 3) = 8*7*6 = 8!/(8-3)! or... How many ways could you put the 8 letters in a hat and draw out a letter, draw out a 2nd letter from that hat that now has only 7 letters and draw out a 3rd letter from the remaining 6 letters and write them down in that order on the board? 8*7*6 is the answer if you DO NOT PUT LETTERS BACK IN THE HAT AFTER DRAWING EACH ONE. 8*8*8 is the answer if you write down the 1st letter, and then throw it back in the hat and mix them up again and then drawn the 2nd letter out, write it down as the 2nd letter, toss it back in the hat, and shake em up again, draw the 3rd letter and write down the 3 letter "word" you got: 3 1st letter 2nd letter 3rd letter = 8*8*8 = 8 PPP is one possible way this could have come out, right? It is more complicated if you draw letters out of the hat and then replace the letter after each time and write down the set of letters you got after 3 drawings. ORDER IS NOT IMPORTANT. It would be C(8, 3) = (8*7*6)/3! = 8*7 = 56 ways or possible sets of 3 letters you could get, right? Very easy, right? WRONG, if we put the letter back in the hat and are always drawing from a hat consisting of 8 letters and say, sorry, if you draw a letter you already had, you do NOT get to draw again. It is MORE COMPLICATED than just C(8, 3) if that is the case. We are playing the following game: Draw a letter out of the hat, replace it, shake the hat. Draw another letter out, replace it, shake the hat again. Draw the final letter out of the hat, and you are done. Write down the SET of possibly 3 letters that you saw. -------- How many sets of letters could be obtained this way, IF WE DO NOT ALLOW YOU TO DRAW AGAIN IF YOU REPEATED A LETTER ALREADY DRAWN EARLIER WHEN DOING THE 2nd AND/OR the 3rd DRAW? C(8,3) + C(8,2) + C(8,1) is the answer. 56 + 28 + 8 = 92 is the answer You might draw the P from PANTHERS all three times, for example. You only have one letter. There are C(8,1) = 8 different ways you could draw the same letter 3 times in a row. Don't laugh, as there is actually 8 chances in 92 or about almost a 9% chance that this would happen. You got the same letter 3 times in a row! Or draw P, then H, then H again, and have the two letters PH. There are 28 or C(8, 2) different sets of TWO different letters you might have encountered in your 3 draws from the hat, where you had a repeat once. For an extra exercise, try this: Place the entire set of letters that make up the word DISCRETE into a hat. Recall MISSISSIPPI letters consist of only the 4 letters M I S P So DISCRET is the entire set of ALL 7 letters for DISCRETE. What is the probability that you get 3 different letters if you randomly draw from the hat --------- of ALL 7 randomly scrambled letters 3 different times? The answer is surprising! :-) Mark