1. Arranging Letters of Words
Many arrangement problems come from rearranging letters of words. Depending on whether letters repeat or not, we choose different formulas.
1.1. All Letters Distinct
If all letters are different, the number of arrangements is simply:
\( n! \)
Example: 'CAT' → 3 distinct letters → \(3! = 6\) arrangements.
1.2. Words with Repeated Letters
If letters repeat, swapping the identical letters does not create new arrangements. So we divide by the factorials of repeated letters.
Number of arrangements:
\( \frac{n!}{p_1! \cdot p_2! \cdots} \)
Example: 'LEVEL' → L appears 2 times, E appears 2 times.
Arrangements = \( \frac{5!}{2! \cdot 2!} \)
2. Forming Numbers Using Digits
Digit-based problems are very common. Depending on repetition rules and conditions, we use different counting methods.
2.1. Numbers Without Repetition
If digits cannot repeat, we reduce choices after each selection.
Example: Form 3-digit numbers using 1, 2, 3, 4 without repetition.
Total = \( ^4P_3 = 24 \)
2.2. Numbers With Repetition
If digits can repeat, each position has the same number of choices.
Example: 3-digit numbers using 0–9 → each digit has 10 options.
Total = \(10^3 = 1000\)
2.3. Numbers With Conditions (Odd, Greater Than a Value, etc.)
Sometimes extra rules apply, such as the number must be odd or must be greater than a given value.
Example: 3-digit odd numbers → last digit must be odd, so choose that first, then fill remaining places.
We break the problem into steps and apply multiplication or case-wise addition.
3. Seating Arrangements
Seating arrangement problems involve placing people in rows or circles. Order matters, so permutations are used.
3.1. Linear Arrangements
Arranging \(n\) people in a row → \(n!\) ways.
Example: 5 friends in a row → \(5! = 120\) ways.
3.2. Circular Arrangements
For circular arrangements, one position is fixed to avoid counting rotations as different.
Number of arrangements = \((n-1)!\)
Example: 6 people around a table → \(5!\) ways.
3.3. Special Conditions (Together, Not Together)
Sometimes conditions are added:
- Together: Treat the pair/group as a single block, arrange the block, then arrange inside.
- Not together: Use total arrangements minus the number of arrangements where they are together.
These problems combine permutations with logical case handling.
4. Committee and Team Selection
Team selection focuses on choosing people, so combinations are used since order does not matter.
4.1. Choosing Members From a Larger Group
Example: Choose 3 members from 10.
Total ways = \( ^{10}C_3 \)
4.2. Selections With Conditions (Must Include / Exclude)
Some problems mention that certain people must be included or cannot be included.
Example: Choose 4 members from 8, but one particular person must be included. First include that person, then choose the remaining 3 from the other 7.
We often split into cases when more than one condition applies.
5. Applying P&C in Simple Probability
P&C methods help count total outcomes and favourable outcomes in probability questions.
5.1. Counting Total Outcomes
Total outcomes are counted first using permutations or combinations, depending on whether order matters.
Example: Number of 2-letter codes from A, B, C, D with repetition → \(4^2 = 16\)
5.2. Counting Favourable Outcomes
Next, we count how many of these outcomes satisfy the given condition.
Example: Codes starting with A → 4 options → favourable = 4.
Probability = favourable outcomes ÷ total outcomes.