Expert: Paul Klarreich - 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
Questioner:   Jon
Category:  Calculus
Private:  No
 
Subject:  Maths
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
..........................................
Hi, Jon,

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


0,2,5,8 you say?  Well, an integer will usually not be written with a leading zero, but although we would not WRITE 0028 that way, it still is an integer.  However, you did say: 1000 to 9999,

So: (1) reps not allowed:  When you do 'no reps' the choices usually go down after you make them.  As in:

First digit:  3 choices. (can't use zero)
Second :  3, since you can't use the first again, but you have your zero back.
Third : 2
Fourth : 1

Total:  3*3*2*1 = 3 * 3! = 18.

(ii) Allowed:  Now they don't go down:

First digit:  3 choices. (can't use zero)
Second :  4, since you have your zero back.
Third : 4
Fourth : 4

Total:  3 * 4^3 = 192,
................................
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.

I think this is  10!/4!.  It's the same as 1(i) -- permutation without reps.

This is  10 9 8 7 6 5  [left out the *'s]

.................................

3. Calculate the number of arrangements of the letters of the word INCLUDE if
(i) all of the consonants are together  

That's NCLD in 4! permutations, which can occur in the 1st, 2nd, 3rd, or 4th spot.
So the vowels, IUE, in 3! perm's, will be interrupted four ways.  For example, take

UIE, now put your NCLD in: the possible spots:

NCLDUIE
UNCLDIE
UINCLDE
UIENCLD

So that's 4 4! 3!
..........................
(ii)no two consonants are together

You must have CVCVCVC, right?  C=cons, V= vowel.

Looks like  4! 3!

(iii)each arrangement begins with a consonant and ends with a vowel

You must have  CxxxxxV.

Four choices for C, three for V, then 5! ways to arrange the x's.

N = 4 5! 3


P.S. Send this stuff to Advanced Math, not Calculus.