EXTRA EXERCISE: PERMUTATIONS AND COMBINATIONS
1. All the letters of the word PHOTOSYNTHESIS are to be arranged in a row.
(a) Find the number of different words that can be formed.
(b) How many of the words formed in (a) begin with the letter H and end with the letter
N?.
(c) Find the number of different ways of arranging the letters such that all the letters S
are separated from each other.
2. Nadijah has 5 red rambutans and 4 yellow rambutans. Find the number of ways in which
seven rambutans can be arranged in a row.
3. (a) How many arrangements can be made by using all the letters of the word
OMNIVOROUS?.
(b) If two letters from the word OMNIVOROUS are used, find the number of possible
arrangements.
(c) How many different ways of arranging all the letters of the word VIRUS which begin
with the letter I or U?.
4. A group of cadets consisting of 3 males and 6 females are to be seated in a row. How
many ways can it be done if all the female cadets are to sit together?.
5. A committee of 3 members is to be chosen from 5 married couples. Find the number of
different committees that can be formed if
(a) All the members are men.
(b) No husband and wife can be included in the same committee.
6. A swimming club comprises of 10 members is to be formed from 7 males and 6 females.
How many clubs can be formed if
(a) there is no restriction on the gender?.
(b) the number of males must be greater than the females?.
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7. The diagram above shows 12 number cards. In how many ways can
(a) 7 cards be selected such that 4 cards are labeled with even numbers and the
remaining cards are labeled with odd numbers?.
(b) 9 cards be arranged in a row such that the first 5 cards are labeled with even
numbers and the remaining cards are labeled with odd numbers?.
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8. An expedition team to Mount Kinabalu consists of 8 men and 4 women.
(a) In how many ways can the team be arranged in a row if all the women must be
together?.
(b) Find the number of ways that a team of 6 members can be formed if the team
consists of
(i)
4 men and 2 women
(ii)
at least 4 men.
9. (a) How many words consisting of three letters that can be formed from the word
SYNDROME if
(i)
repetition of letters are not allowed.
(ii)
each letter can only be used once and no word begins with the letter Y?.
(b) How many ways can two letters be chosen from the word SYN and three letters
from the word DROME?.
10. How many ways can all the letters of the words PETALING JAYA be arranged
(a) if the first letter is P and the last letter is A?.
(b) all the letters A must be together?.
(c) the first letter is a consonant and the last letter is a vowel?.
11. In a public speaking competition, 7 out of the 11 contestants are females. Find the
number of ways of choosing
(a) the first place, the second place and third place winners.
(b) four winners consisting of at least two females.
(c) two females and one male winners for the first place, the second place and the third
place.
12. A group of 12 environmentalists comprising of 7 engineers and 5 biologists are to be
selected to form a committee. Find the number of ways of
(a) forming a committee of 5 members consisting of at least 3 engineers,
(b) arranging all the 12 environmentalists in a row such that they are always in the
group of the same expertise,
(c) selecting 3 members for the post of chairman, secretary and treasurer,
(d) forming a committee of 5 members with an engineer as chairman and a biologist as
secretary.
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13. There are 5 men, 8 women and 20 students participating in a social gathering.
(a) In how many ways can a group of 3 men, 5 women and 16 students be formed if a
particular woman and a particular student must be included in the group?.
(b) Twenty five participants are required in an activity. In how many ways can this group
be formed if each group must consist of at least 6 women.
14. A five-digit number is to be formed from the digits 2, 3, 4, 5, 6, 7, 8 and 9 without
repetition. How many
(a) five-digit numbers between 40000 and 70000 can be formed?
(b) five-digit odd numbers greater than 70000 can be formed?
15. (a) At a wedding reception, 4 cars which can carry 2, 4, 4 and 4 passengers respectively
are provided for transporting 14 guests. Find how many ways are possible to
transport all the 14 passengers.
(b) If 4 cars with capacity to transport 3, 4, 4 and 4 passengers respectively are used,
how many ways can this be done?.
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16. The diagram above shows 6 digits and 4 letters. A secret code is to be formed using
those digits and letters. The secret code must consist of 4 digits followed by 2 letters.
How many secret codes can be formed if no digit or letter is repeated in each code.
17. (a) In how many ways can 5 boys and 4 girls be arranged in a row if
(i)
the row begins with a boy and ends with a girl?.
(ii)
the girls must be separated from each other?.
(b) In how many ways can a four-member committee be formed from 5 boys and 4 girls
if the committee must consist of equal number of both genders?.
18. The panel of judges of a logo design competition will determine the winner for the first
place to the fifth place. If 14 females and 6 males participate in the competition, in how
many ways can one select
(a) three female winners for the first place to the third place and two male winners for
the fourth place and the fifth place?.
(b) five winners consisting of three females and two males?.
(c) at least three female winners in the competition?.
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19. In a swimming competition comprising of 12 participants, prizes are awarded to the first
three winners.
(a) Find the possible number of outcomes.
(b) If Ali is a participant, find the number of ways he can win a prize in the competition.
20. In how many ways can 7 men and 5 women be arranged in a row if
(a) there is no restriction?.
(b) the 7 men must be together?.
(c) no women are adjacent?.
21. In how many ways can the word UNUSUAL be arranged if no two U’s are adjacent?.
22. Three delegates are to be chosen from a group of 4 lawyers, an engineer and three
professors. In how many ways can the delegation be chosen if it must include at least
one professor?.
23. Given the digits 2, 3, 4, 5, 6. With no repetition, find the number of
(a) Odd numbers that can be formed.
(b) 5-digit even numbers greater than 30000 that can be formed.
Answer:
1(a) 1816214400 (b) 19958400 (c) 1463616000 2. 91 3(a) 604800 (b) 57 (c) 48 4. 17280
5(a) 10 (b) 80 6(a) 286 (b) 125 7(a) 300 (b) 259200 8(a) 8709120 (b)(i) 420 (ii) 672
9(a)(i) 336 (ii) 294 (b) 30 10(a) 1814400 (b) 3628800 (c) 21168000 11(a) 990 (b) 301 (c) 504
12(a) 546 (b) 1209600 (c) 1320 (d) 4200 13(a) 1356600 (b) 9885975 14(a) 2520 (b) 1200
15(a) 3153150 (b) 15765750 16. 4320 17(a)(i) 100800 (ii) 43200 (b) 60 18(a) 65520
(b) 655200 (c) 1616160 19(a) 1320 (b) 330 20(a) 479001600 (b) 3628800 (c) 3368800
21. 240 22. 46 23(a) 48 (b) 60
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