Counting is at the heart of probability and statistics. This chapter gives us systematic tools — the Fundamental Principle of Counting, Permutations, and Combinations — to count arrangements and selections.
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Fundamental Principle of Counting
Multiplication Principle: If event A can occur in m ways and event B (independent of A) can occur in n ways, then both A and B together can occur in m × n ways.
Addition Principle: If event A can occur in m ways OR event B in n ways (mutually exclusive), then the total is m + n ways.
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Factorial
Key formulas
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Permutations
A permutation is an arrangement of objects in a specific order.
P(n, r) = n!/(n−r)! = number of ways to arrange r objects chosen from n distinct objects.
- Special cases:
- P(n, n) = n! (arrange all n objects)
- Permutations of objects with repetitions: If there are n objects where p are of one type, q of another, etc., then total arrangements = n!/(p! × q! × …)
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Combinations
A combination is a selection of objects where order does not matter.
C(n, r) = n!/(r! × (n−r)!) = number of ways to choose r objects from n distinct objects.
- Key properties:
- C(n, r) = C(n, n−r)
- C(n, 0) = C(n, n) = 1
- C(n, r) + C(n, r−1) = C(n+1, r) (Pascal's identity)
- C(n, 1) = n
Difference between P and C: Permutations count arrangements (order matters); combinations count selections (order does not).
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Worked Examples
In how many ways can 3 prizes be given to 5 students if no student gets more than one prize?
P(5,3) = 5!/(5−3)! = 5!/2! = 120/2 = 60.
How many 3-digit numbers can be formed using digits 1, 2, 3, 4, 5 without repetition?
P(5,3) = 60.
In how many ways can a committee of 3 be chosen from 10 people?
C(10,3) = 10!/(3! × 7!) = (10 × 9 × 8)/(3 × 2 × 1) = 120.
Find the number of arrangements of the letters of the word MISSISSIPPI.
M=1, I=4, S=4, P=2. Total = 11!/(4! × 4! × 2!) = 34650.
How many words can be formed using all letters of PENCIL?
6 distinct letters. Answer = 6! = 720.
From 6 boys and 4 girls, a team of 5 is to be selected with at least 3 girls. Find the number of ways.
Case 1 (3 girls, 2 boys): C(4,3) × C(6,2) = 4 × 15 = 60.
Case 2 (4 girls, 1 boy): C(4,4) × C(6,1) = 1 × 6 = 6.
Total = 60 + 6 = 66.
Prove that C(n, r) = C(n, n−r).
C(n, n−r) = n!/((n−r)! × r!) = n!/(r! × (n−r)!) = C(n,r). Proved.
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Common mistakes
- Confusing permutation and combination: ask "Does order matter?" If yes → P; if no → C.
- Forgetting that 0! = 1.
- Incorrectly computing factorials for large n (simplify by cancelling terms).
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Summary
The multiplication and addition principles are the foundation. Permutations count ordered arrangements (use P(n,r)); combinations count unordered selections (use C(n,r)). Pascal's identity and the complementary property C(n,r) = C(n,n-r) are useful shortcuts.