Math 30 Permutations with Restrictions Worksheet
A. Together, Not together
1. How many arrangements could be made of the following word FATHER if
a. all the vowels must be kept together? (240)
b. the letters T, H, and E cannot be together? (576)
2. Ann, Brian, Colin, Diane and Eric go to watch a movie and sit in 5 adjacent seats. In how
many ways can this be done if
a. Brian sits next to Diane? (48)
b. Ann refuses to sit next to Eric? (72)
c. members of the same gender must be adjacent? (24)
3. A student has 5 different English books and 4 different French books. In how many
ways can these books be arranged on a bookshelf if:
a. the books on a particular subject must appear together. (5760)
b. the English books must appear to the left of the French books. (2880)
A. Together, Not together
4. How many ways can 3 different algebra books, 4 different geometry books and 2 different
calculus books be arranged on a shelf with space for 9 books if the calculus books cannot
be together? (282 240)
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Same order, ascending, and descending order
5. Using the letters of the word HYPERBOLA, how many 9 letter words can be formed if the
order of the letters P, E, and R must be the same. (60 480)
6. How many arrangements are there of the digits 1, 2, 3, 4, 5, if the odd digits must be in
ascending order? (20)
7. Find the number of arrangements of the letters in the word ANSWER that have the
three letters A, N, and S adjacent and in the order ANS? (24)
8. How many different whole numbers can be obtained by arranging the digits 1, 2, 3, … 9
all at a time, if the even numbers must always remain in descending order and
the odd digits likewise. (126)
9. How many arrangements can be made using the letters of EQUATIONS provided that
the vowels always remain in the same order?
(3024)
Positional Restrictions
10. Find the number of different arrangements of the letters in the word CHANGES under
each condition:
a. without restrictions (7!)
b. P
6-letter arrangements ( 7 6 ) or 7 6 5 4 3 2
c. starts with an ‘S’ d. has a vowel in the middle ( 6!)
( 6!)
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e. ends with a consonant f. first two letters are vowels (
6! 5 ) ( 2 5!)
g. position of the vowels do not change 5!
h ‘S’ must be on either end i. ends with AN
(2 6! ) (120)
j. begins with a vowel and ends with a consonant
(1200)
Positional Restriction
11. Find the number of 4 digit natural numbers that can be formed that contain the digit 3.
(3 168)
Alternating
12. In how many ways can 4 adults and 5 children be arranged in a single line if the
children and adults are alternated?
(2880)
13. In how many ways can 5 boys and 5 girls be arranged in a row
a. if the boys and girls are alternated? (28 800)
b. if the boys and girls alternate but a boy is at the beginning of the row? (14 400)
At least & At most
13. Find the number of 4-letter words that can be formed from the letters of the word
FACETIOUS, if at least one consonant must be used. (2
904)
14. Given 5 flags of different colors, how many different signals can be made by arranging
the flags on a vertical mast, if at least 3 flags must be used for each signal?
(300)
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15. How many signals, using 4 different lights, can be made using, at most two of them in a
row? (16)
Separate Restriction
16. In how many ways can the letters of the word CHAPTER be arranged if the letters T, E,
and R are separated by other letters?
(1 440)
17. In how many ways can the letters of the word JASPER be arranged if the vowels must be
separated by other letters? (480)
Mixed Restriction
18. In how many ways can all the letters of the word MONTREAL be arranged if the letter
R, E, A, and L must appear as the word REAL in all arrangements? (120)
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