Discrete Mathematical
Structures
Lecture 06
Techniques of Counting
• The number of possible outcomes of a particular event or the number of
elements in a set.
• Such sophisticated counting is sometimes called combinatorial analysis.
• It includes the study of permutations and combinations.
• Sum Rule Principle:
𝑆𝑢𝑝𝑝𝑜𝑠𝑒 𝑠𝑜𝑚𝑒 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟 𝑖𝑛 𝑚 𝑤𝑎𝑦𝑠 𝑎𝑛𝑑 𝑎 𝑠𝑒𝑐𝑜𝑛𝑑 𝑒𝑣𝑒𝑛𝑡 𝐹
𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟 𝑖𝑛 𝑛 𝑤𝑎𝑦𝑠, 𝑎𝑛𝑑 𝑠𝑢𝑝𝑝𝑜𝑠𝑒 𝑏𝑜𝑡ℎ 𝑒𝑣𝑒𝑛𝑡𝑠 𝑐𝑎𝑛𝑛𝑜𝑡 𝑜𝑐𝑐𝑢𝑟 𝑠𝑖𝑚𝑢𝑙𝑡𝑎𝑛𝑒𝑜𝑢𝑠𝑙𝑦.
𝑇ℎ𝑒𝑛 𝐸 𝑜𝑟 𝐹 𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟 𝑖𝑛 𝑚 + 𝑛 𝑤𝑎𝑦𝑠.
• Product Rule Principle:
𝑆𝑢𝑝𝑝𝑜𝑠𝑒 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎𝑛 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑤ℎ𝑖𝑐ℎ 𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟 𝑖𝑛 𝑚 𝑤𝑎𝑦𝑠 𝑎𝑛𝑑, 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑓
𝑡ℎ𝑖𝑠 𝑒𝑣𝑒𝑛𝑡, 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑠𝑒𝑐𝑜𝑛𝑑 𝑒𝑣𝑒𝑛𝑡 𝐹 𝑤ℎ𝑖𝑐ℎ 𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟 𝑖𝑛 𝑛 𝑤𝑎𝑦𝑠.
𝑇ℎ𝑒𝑛 𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛𝑠 𝑜𝑓 𝐸 𝑎𝑛𝑑 𝐹 𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟 𝑖𝑛 𝑚𝑛 𝑤𝑎𝑦𝑠.
Techniques of Counting
Example: Let there be 2 cats and 3 dogs. If you can add them together in a room, how many
Animals would there be in room?
2+3=5
In how many ways we can arrange pairs:
{𝐶1 , 𝐶2 , 𝐷1 , 𝐷2 , 𝐷3 } => {𝐶1 𝐷1 , 𝐶1 𝐷2 , 𝐶1 𝐷3 , 𝐶2 𝐷1 , 𝐶2 𝐷2 , 𝐶2 𝐷3 }
Mathematical functions
Mathematical functions
Binomial Coefficients and Pascal’s Triangle
Permutations
Permutations
Permutations with Repetitions
Example
Ordered Samples
Ordered Samples
Example
Combinations
Example
Combination
The pigeonhole principle
Suppose a department contains 13 professors, then two of the professors were born in the
same month
Example
Practice Question