Tuples & Permutations
Counting ordered sequences with and without repetition
🔁 Tuples (With Repetition)
Definition
A tuple is an ordered sequence of length \(k\) where symbols are chosen from a set of \(n\) elements, allowing repetitions.
Formula: \(n^k\)
Interactive: Password Search Space
Type a password to see how many possibilities exist for that length!
Example: Vehicle Number Plates
Consider a simplified Indian number plate format:
- • State (2 letters): \(26 \times 26\) options
- • District (2 digits): \(10 \times 10\) options
- • Series (2 letters): \(26 \times 26\) options
- • Number (4 digits): \(10^4\) options
Total Vehicles = \(26^2 \times 10^2 \times 26^2 \times 10^4 \approx 4.5 \times 10^9\)
🔀 Permutations (No Repetition)
Definition
A \(k\)-permutation is an ordered sequence of length \(k\) from \(n\) symbols, where no symbol is repeated.
Formula: \(P(n, k) = \frac{n!}{(n-k)!}\)
Interactive: Filling Slots
We have 5 distinct items (A, B, C, D, E). How many ways to fill 3 slots?
Total = \(5 \times 4 \times 3 = 60\)
Notice how the number of options decreases by 1 for each slot!
📝 Test Your Understanding
Question 1:
How many 3-digit PIN codes can be formed using digits 0-9 (repetition allowed)?
Question 2:
In a race with 8 runners, how many ways can the Gold, Silver, and Bronze medals be awarded?