Ch.17 Permutation and Combination
1 Direct multiplication
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Example:
If three people A, B and C go to a 3 stories
hotel, how many arrangements are there such
that A and B would not be on the same floor?
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 =3×2×3=18
For A is on 1/F, B must be on 2/F or 3/F, vice versa.
2 Permutation
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𝑃𝑟𝑛 =
𝑛!
(𝑛 − 𝑟 )!
Example:
How many ways can Andy, Bob, Cathy be lined up?
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 = 𝑃33 = 6
Note that
There are 3 people
The “capacity” of the line is 3
We may see it as: 𝑃33
Note that
Unlike combination, permutation includes arrangement.
For example, “AndyBobCathy” and “AndyCathy Bob” are different.
3 Combination
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𝐶𝑟𝑛 =
𝑛!
(𝑛 − 𝑟 )! 𝑟!
Example:
How many ways can Andy, Bob, Cathy form a team of 2?
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 = 𝐶23 = 3
Note that
We may see it as: 𝐶23
There are 3 people
The “capacity” of the team is 2
Note that
Unlike permutation, combination excludes arrangement.
For example, “AndyBob” and “BobAndy” are the same.
4 Common question type
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Digits
Example:
How many 3-digit numbers can 0, 1 and 2 form?
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 = 𝑃33 − 𝑃22 = 4
012
021
102
120
201
210
012
021
Note that
Digits cannot start with 0
Letters and Words
Example:
How many different words can B, A, L, L form?
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 = 𝐶24 × 2 = 12
Consider the two “L” first:
There are 4 spaces to place 2 “L”
Lining up
Example:
If Andy, Bob, Cathy and Daniel line up, find the number of
ways such that Daniel is behind Cathy
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 = 𝑃44 ÷ 2 = 12
Note that
Daniel is either behind or in front of Cathy
Example:
There are 2 boys and 3 girls lining up. Find the number of
ways such that the boys are standing next to each other.
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 = 4 × 𝑃22 × 𝑃33 = 48
𝑃22
𝑃33
_B _B _G _G _G
The “box” can be in
4 different position
of the 5 spaces
Separation
Group with less individuals
Group with more individuals
Example:
There are 4 boys and 3 girls lining up. Find the number of
ways such that no boys are next to each other.
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 = 𝑃44 × 𝑃33 = 144
5 Complementary Cases
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𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒘𝒂𝒚𝒔 𝒇𝒐𝒓 𝒆𝒗𝒆𝒏𝒕 𝑨 𝒏𝒐𝒕 𝒕𝒐 𝒉𝒂𝒑𝒑𝒆𝒏
= 𝑨𝒍𝒍 𝒐𝒖𝒕𝒄𝒐𝒎𝒆 − 𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒘𝒂𝒚𝒔 𝒇𝒐𝒓 𝒆𝒗𝒆𝒏𝒕 𝑨 𝒕𝒐 𝒉𝒂𝒑𝒑𝒆𝒏
Example:
Find the number of ways such that there are at least 1 head in
5 tosses of a fair coin.
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠
= 𝐴𝑙𝑙 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 − 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑜𝑓 𝑔𝑒𝑡𝑡𝑖𝑛𝑔 𝑎𝑙𝑙 𝑡𝑎𝑖𝑙
= 25 − 1 = 31