Permutations and Combinations A/AS level part 2
Permutations and Combinations: A Comprehensive Summary
1. Permutations
Permutations are arrangements of objects where order matters.
Formula:
For n distinct objects, the number of permutations is:
P(n) = n!For r objects selected from n distinct objects:
P(n,r) = n! / (n-r)!Example:
Arranging the letters in "REQUIREMENT":
11! / (2!3!) = 3,326,400(There are 11 letters, with 2 Rs and 3 Es repeated)
2. Combinations
Combinations are selections of objects where order doesn't matter.
Formula:
Selecting r objects from n distinct objects:
C(n,r) = n! / (r!(n-r)!)Example:
Selecting 4 toys from 9 toys:
C(9,4) = 9! / (4!5!) = 1263. Conditional Probability in Selections
When selecting objects with conditions, break down the problem into cases:
Example:
Selecting 6 people from 8 men and 4 women, with at least twice as many men as women:
- Case 1: 4 men, 2 women: C(8,4) * C(4,2)
- Case 2: 5 men, 1 woman: C(8,5) * C(4,1)
- Case 3: 6 men, 0 women: C(8,6)
Total = Sum of all cases
4. Arrangement with Restrictions
When arranging objects with restrictions, consider the restrictions as blocks:
Example:
Arranging "SUMMERTIME" with Es together:
8! * 2! = 80,640(Treat "EE" as one unit, then arrange 8 units, multiply by ways to arrange "EE")
5. Inclusion-Exclusion Principle
For selecting objects with "at least" conditions, use this principle:
P(A or B) = P(A) + P(B) - P(A and B)Example:
Selecting 3 letters from "CAMERAMAN" with at least one M:
Total - (selections without M) = C(9,3) - C(7,3)Conclusion
These principles form the foundation for solving complex permutation and combination problems. Always identify whether order matters (permutation) or doesn't (combination), and break down complex problems into simpler cases when necessary.
Exercise:
Questions from 9709 Past Papers
85. 9709_s21_qp_51 Q: 1
A bag contains 12 marbles, each of a different size. 8 of the marbles are red and 4 of the marbles are blue.
How many different selections of 5 marbles contain at least 4 marbles of the same colour?
86. 9709_s21_qp_53 Q: 6
(a) How many different arrangements are there of the 11 letters in the word REQUIREMENT?
(b) How many different arrangements are there of the 11 letters in the word REQUIREMENT in which the two Rs are together and the three Es are together?
(c) How many different arrangements are there of the 11 letters in the word REQUIREMENT in which there are exactly three letters between the two Rs?
(d) Five of the 11 letters in the word REQUIREMENT are selected. How many possible selections contain at least two Es and at least one R?
87. 9709_w21_qp_52 Q: 2
A group of 6 people is to be chosen from 4 men and 11 women.
(a) In how many different ways can a group of 6 be chosen if it must contain exactly 1 man?
(b) Two of the 11 women are sisters Jane and Kate. In how many different ways can a group of 6 be chosen if Jane and Kate cannot both be in the group?
88. 9709_w21_qp_52 Q: 4
(a) In how many different ways can the 9 letters of the word TELESCOPE be arranged?
(b) In how many different ways can the 9 letters of the word TELESCOPE be arranged so that there are exactly two letters between the T and the C?
89. 9709_w21_qp_53 Q: 1
The 26 members of the local sports club include Mr and Mrs Khan and their son Abad. The club is holding a party to celebrate Abad's birthday, but there is only room for 20 people to attend.
In how many ways can the 20 people be chosen from the 26 members of the club, given that Mr and Mrs Khan and Abad must be included?
90. 9709_m20_qp_52 Q: 1
The 40 members of a club include Ranuf and Saed. All 40 members will travel to a concert. 35 members will travel in a coach and the other 5 will travel in a car. Ranuf will be in the coach and Saed will be in the car.
In how many ways can the members who will travel in the coach be chosen?
91. 9709_m20_qp_52 Q: 4
Richard has 3 blue candles, 2 red candles and 6 green candles. The candles are identical apart from their colours. He arranges the 11 candles in a line.
(a) Find the number of different arrangements of the 11 candles if there is a red candle at each end.
(b) Find the number of different arrangements of the 11 candles if all the blue candles are together and the red candles are not together.
92. 9709_s20_qp_51 Q: 2
(a) Find the number of different arrangements that can be made from the 9 letters of the word JEWELLERY in which the three Es are together and the two Ls are together.
(b) Find the number of different arrangements that can be made from the 9 letters of the word JEWELLERY in which the two Ls are not next to each other.
93. 9709_s20_qp_51 Q: 4
In a music competition, there are 8 pianists, 4 guitarists and 6 violinists. 7 of these musicians will be selected to go through to the final.
How many different selections of 7 finalists can be made if there must be at least 2 pianists, at least 1 guitarist and more violinists than guitarists?
94. 9709_s20_qp_52 Q: 6
(a) Find the number of different ways in which the 10 letters of the word SUMMERTIME can be arranged so that there is an E at the beginning and an E at the end.
(b) Find the number of different ways in which the 10 letters of the word SUMMERTIME can be arranged so that the Es are not together.
(c) Four letters are selected from the 10 letters of the word SUMMERTIME. Find the number of different selections if the four letters include at least one M and exactly one E.
95. 9709_w20_qp_53 Q: 3
A committee of 6 people is to be chosen from 9 women and 5 men.
(a) Find the number of ways in which the 6 people can be chosen if there must be more women than men on the committee.
The 9 women and 5 men include a sister and brother.
(b) Find the number of ways in which the committee can be chosen if the sister and brother cannot both be on the committee.
96. 9709_m19_qp_62 Q: 7
Find the number of different arrangements that can be made of all 9 letters in the word CAMERAMAN in each of the following cases.
(i) There are no restrictions.
(ii) The As occupy the 1st, 5th and 9th positions.
(iii) There is exactly one letter between the Ms.
Three letters are selected from the 9 letters of the word CAMERAMAN.
(iv) Find the number of different selections if the three letters include exactly one M and exactly one A.
(v) Find the number of different selections if the three letters include at least one M.
97. 9709_s19_qp_61 Q: 8
Freddie has 6 toy cars and 3 toy buses, all different. He chooses 4 toys to take on holiday with him.
(i) In how many different ways can Freddie choose 4 toys?
(ii) How many of these choices will include both his favourite car and his favourite bus?
Freddie arranges these 9 toys in a line.
(iii) Find the number of possible arrangements if the buses are all next to each other.
(iv) Find the number of possible arrangements if there is a car at each end of the line and no buses are next to each other.
98. 9709_s19_qp_62 Q: 7
(a) A group of 6 teenagers go boating. There are three boats available. One boat has room for 3 people, one has room for 2 people and one has room for 1 person. Find the number of different ways the group of 6 teenagers can be divided between the three boats.
(b) Find the number of different 7-digit numbers which can be formed from the seven digits 2, 2, 3, 7, 7, 7, 8 in each of the following cases.
(i) The odd digits are together and the even digits are together.
(ii) The 2s are not together.
99. 9709_s19_qp_63 Q: 3
Mr and Mrs Keene and their 5 children all go to watch a football match, together with their friends Mr and Mrs Uzuma and their 2 children. Find the number of ways in which all 11 people can line up at the entrance in each of the following cases.
100. 9709_s19_qp_63 Q: 4
(i) Find the number of ways a committee of 6 people can be chosen from 8 men and 4 women if there must be at least twice as many men as there are women on the committee.
(ii) Find the number of ways a committee of 6 people can be chosen from 8 men and 4 women if 2 particular men refuse to be on the committee together.
