Expert: Scotto - 2/3/2008

Question
hi.these are a few questions (Permutations) I have attempted.Could you please guide me if I am going wrong.Thank You

1. Calculate the number of integers between 1000 and 9999 which can be formed from the digits 0,2,5 and 8 (i)if repetitions are not allowed(ii)if repetitions are allowed

ANS: (i)P= 4!/3!=4   
    
     (ii)P= 4!
(i am not 100% sure how to handle the issue of repetitions with or w/out)

2. A shelf will only hold 6 books. Given that 10 different books are available,find the number of different arrangements which can be made to fill the shelf.

ANS: 10!/6!=5040

3. Calculate the number of arrangements of the letters of the word INCLUDE if
(i) all of the consonants are together  
(ii)no two consonants are together
(iii)each arrangement begins with a consonant and ends with a vowel

ANS:(i) 7!/4!3!
   (ii)7!/3!
   (iii)(7!/5!) * 3=126

Answer
1. i) If repetions are not allowed, you have n items in any order, which is n! except for the fact that 1/4 of the choices must be excluded since they start with 0.
ii) If repetitions are allowed, it's just 3*4^3 since the first digit still must not be 0.

2. There are 10 ways of selecting the first book, 9 ways of the second, .... , 5 ways of selecting the sixth book.  This amounts to 10!/4!.

3. i) C=consonants, V=vowels; 4*C!V! (4* for consonants/vowels or vowels/consonants) - that answer does not look right per the problem stated.  Here is the number of arrangements of the consonants:
1 N C D L
2 N C L D
3 N D C L
4 N D L C
5 N L C D
6 N L D C
7 C N D L
8 C N L D
9 C D N L
10 C D L N
11 C L N D
12 C L D N
13 D N C L
14 D N L C
15 D C N L
16 D C L N
17 D L N C
18 D L C N
19 L N C D
20 L N D C
21 L C N D
22 L C D N
23 L D N C
24 L D C N
They can be arrnaged with 0, 1, 2, or 3 vowels to the left (which makes the answer *4) and the vowels can be arranged in 3!=6 ways.

ii) The consonants must be 1st, 3rd, 5th, and 7th.  As was shown in the last problem, there are 4! ways to arrange the consonants and 3! ways to arrange the vowels.

iii) Look at all of the possible combinations - 7!.  4/7 of these combinations start with a consonant.  Of the ones that are left, there are 3 consonants and 3 vowels in the rest of the word, so 3/6, or 1/2 of the arrangements remaining end in a vowel.