HSSLIVE Plus One Maths Chapter 7: Permutation and Combinations Notes

This chapter introduces the fundamental principles of counting that underlie probability theory and discrete mathematics. Students will learn to determine the number of possible arrangements (permutations) and selections (combinations) of objects under various conditions. These powerful combinatorial tools provide students with systematic ways to analyze complex counting problems, optimize decision-making processes, and solve probability questions without enumerating all possibilities—essential skills for data science, cryptography, and algorithm design.

Chapter 7: Permutation and Combinations

Permutations and combinations are ways of selecting and arranging objects from a collection.

Fundamental Counting Principle:

If an event can occur in m ways, and another independent event can occur in n ways, then the total number of ways both events can occur is m × n.

Factorial Notation:

n! = n × (n-1) × (n-2) × … × 3 × 2 × 1 Special cases: 0! = 1 and 1! = 1

Permutations:

Permutation refers to an arrangement of objects where order matters.

• Permutation of n different objects taken all at a time: P(n,n) = nPn = n!
• Permutation of n different objects taken r at a time: P(n,r) = nPr = n!/(n-r)!
• Permutation with repetition: If there are n objects with n₁ of one kind, n₂ of second kind, …, nk of kth kind, then: Number of permutations = n!/(n₁! × n₂! × … × nk!)

Combinations:

Combination refers to selection of objects where order doesn’t matter.

• Combination of n different objects taken r at a time: C(n,r) = nCr = n!/[r! × (n-r)!]

Properties of Combinations:

1. nCr = nC(n-r)
2. nC0 = nCn = 1
3. nCr + nC(r-1) = (n+1)Cr

Applications:

• Team selection
• Sampling methods
• Probability calculations
• Committee formation