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Disc-a1 - 50 CDs(a How many ways can you load the player so...

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1. How many four letter words (including nonsensical words) are there that do not have two consecutive letters the same? How many are there that do have two consecutive letters the same ? We can choose the °rst letter in 26 ways. Having chosen the °rst letter, we can choose the second letter in 25 ways because it has to be di/erent from the °rst letter. Having chosen the °rst two letters, we can choose the third letter in any of 25 ways because it has to be di/erent from the second letter (but not the °rst). Having chosen the °rst three letters, we can choose the fourth letter in any of 25 ways because it has to be di/erent from the third letter. So the number of four letter words that do not have two consecutive letters the same is 26 ° 25 ° 25 ° 25 = 406 ; 250 . The total number of four letter words is 26 ° 26 ° 26 ° 26 = 456 ; 976 , so the number of them that have two consecutive letters the same is 26 4 ± 26 ° 25 3 = 456 ; 976 ± 406 ; 250 = 50 ; 726 : 2. A CD player has 6 trays for CDs. You have
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Unformatted text preview: 50 CDs . (a) How many ways can you load the player so that each tray contains a CD ? You can put any of the 50 CDs in the &rst tray, any of the 49 re-maining CDs in the second tray, and so on. So the number of ways you can load the player is 50 & 49 & 48 & 47 & 46 & 45 = 11 ; 441 ; 304 ; 000 . (b) How many ways can you load the player so that exactly one tray contains a CD ? You have to pick one of the 50 CDs to load and one of the 6 trays to put it in. The number of ways you can do this is 50 & 6 = 300 . 3. Calculate the following (a) 100! 98! 100! 98! = 100 & 99 & 98 & 97 & & & & & 1 98 & 97 & & & & & 1 = 100 & 99 = 9900 (b) (51) 3 (51) 3 = 51 & 50 & 49 = 124 ; 950 (c) Q 6 k =3 (2 k ± 1) Q 6 k =3 (2 k ± 1) = 5 & 7 & 9 & 11 = 3465 (d) Q 100 k =1 k k +1 Q 100 k =1 k k +1 = 1 2 & 2 3 & 3 4 & 4 5 & & & & & 98 99 & 99 100 & 100 101 = 1 101...
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