Permutation-Based Probability
When managing large groups or sequential arrangements, listing out outcomes manually becomes impractical. Instead, we use combinations and permutations to count possibilities efficiently.
Fundamental Principles
Combination Formula nCr
Used when selection order does not matter: nCr = n! / [r!(n - r)!], where 'n' is the total pool and 'r' is the number of items being chosen.
Permutation Formula nPr
Used when selection order matters: nPr = n! / (n - r)!
Essential Formulation Tips
- Use Combinations (nCr) for committee selections or drawing groups of balls, where order doesn't matter. Use Permutations (nPr) for seating arrangements or word formations, where order does matter.
Shortcut Execution Techniques
- When an arrangement problem requires certain items to always stay next to each other, tie those items together and treat them as a single structural block during calculations.
Contextual Inquiries (FAQs)
Q: What does the exclamation mark symbol (!) mean in these formulas?
A:
Example Breakdown: Calculating Group Selection Combinations
Standard full-scale combination architecture template used in advanced exam papers.Step 1: Calculate total possible ways to choose 3 people out of 9: 9C3 = (9×8×7)/(3×2×1) = 84 ways.
Step 2: Calculate ways to choose 2 men from 5: 5C2 = 10 ways.
Step 3: Calculate ways to choose 1 woman from 4: 4C1 = 4 ways.
Step 4: Multiply favorable choices: 10 × 4 = 40 ways.
Step 5: Form the final fraction: 40 / 84 = 10 / 21.
Permutation-Based Probability Practice Set 1
10 questions evaluating committee assignments, ball drawings, and letter arrangements using nCr counting rules.
Q1. A bag holds 6 white and 4 black marbles. If 2 marbles are drawn at random, find the probability that both are white.
Q2. A box contains 4 red, 5 green, and 3 blue toys. If 2 toys are drawn at random, find the probability that both are green.
Q3. The letters of the word 'LEAD' are arranged in a random order. Find the probability that the vowels sit next to each other.
Q4. Out of 5 engineers and 3 designers, a committee of 4 people is being formed. Find the probability that it includes exactly 2 engineers and 2 designers.
Q5. A bag contains 6 black and 4 red balls. Three balls are drawn at random. What is the probability that 2 are black and 1 is red?
Q6. If 3 cards are drawn at random from a standard deck of 52 cards, which expression represents the total number of combinations in the sample space?
Q7. Four items are drawn together from a box holding 12 sound units. If 3 units are defective, find the probability that none of the 4 drawn items are defective.
Q8. The letters of the word 'CAT' are rearranged randomly. What is the probability that the word starts with a consonant?
Q9. A basket holds 5 red, 3 green, and 2 yellow apples. If 3 apples are pulled at random, find the probability that all 3 are red.
Q10. Two distinct people are selected at random from a group of 4 men and 2 women. Find the probability that at least one woman is chosen.