Data Handling
Permutations, combinations & probability
Count outcomes with permutations and combinations, then compute probabilities in finite sample spaces.
Counting methods make probability efficient when listing all outcomes is impractical.
Factorial
\[ n!=n(n-1)(n-2)\cdots 1,\quad 0!=1. \]
Permutations (order matters)
\[ {}_nP_r=\frac{n!}{(n-r)!}. \]
Use for arrangements where position/order is important.
Combinations (order not important)
\[ {}_nC_r=\frac{n!}{r!(n-r)!}. \]
Use for selections/groups.
Probability with counting
\[ P(E)=\frac{\text{count of favorable outcomes}}{\text{count of all outcomes}}. \]
With/without replacement
- With replacement: probabilities stay same each draw.
- Without replacement: probabilities change each draw.
Exam strategy
- First decide: arrangement or selection?
- Check if repetitions are allowed.
- Use structured counting (tree/cases) for restrictions.
- Simplify probability fractions at end.
Checkpoints
Compute \({}_7P_3\).
Answer: \(7\times6\times5=210\).
Compute \({}_8C_2\).
Answer: \(\frac{8\cdot7}{2}=28\).
How many 3-letter arrangements from A,B,C,D (no repeats)?
Answer: \({}_4P_3=24\).
How many ways to choose 3 students from 10?
Answer: \({}_{10}C_3=120\).
A 5-card hand from 52 cards: probability all hearts.
Answer: \(\frac{{}_{13}C_5}{{}_{52}C_5}\).
Last modified: