Permutations and combinations: arrangements, selections

Permutations and Combinations: Arrangements & Selections 📚

1️⃣ Arrangements (Permutations) 🔄

When the order matters, we talk about permutations. Think of arranging books on a shelf or lining up for a photo.

• Formula for arranging all n items:
  $P(n)=n!$
• Formula for arranging r out of n items (order matters):
  $P(n,r)=\dfrac{n!}{(n-r)!}$
• Example: 3 different coloured balls (Red, Blue, Green) in a line:
  $P(3)=3!=6$ ways: RBG, RGB, BRG, BGR, GRB, GBR.
• With repeated items: If 2 balls are the same colour, divide by the factorial of the repeats:
  $\dfrac{n!}{k_1!k_2!\dots}$

2️⃣ Selections (Combinations) 📋

When the order does not matter, we use combinations. Imagine picking a team of 4 from 10 students.

• Formula for choosing r out of n (no order):
  $C(n,r)=\binom{n}{r}=\dfrac{n!}{r!(n-r)!}$
• Example: Choose 2 out of 5 friends:
  $\binom{5}{2}=10$ possible pairs.
• With identical items (stars and bars): For selecting r items from n types with unlimited supply:
  $\binom{n+r-1}{r}$
• Practice Tip: If you can count the arrangements and then divide by the number of ways to reorder the chosen items, you get the combinations.

3️⃣ Quick Reference Table 📊

4️⃣ Quick Practice Problems 🧩

1. How many ways can 5 different shirts be worn in a row if you only wear 3?
   Answer: $P(5,3)=\dfrac{5!}{2!}=60$
2. From 12 students, how many 4‑person teams can be formed?
   Answer: $\binom{12}{4}=495$
3. Choose 2 sweets from 3 types (apple, banana, cherry) with unlimited supply. How many combinations?
   Answer: $\binom{3+2-1}{2}=6$
4. Arrange the letters of the word "LEVEL". How many distinct arrangements?
   Answer: $\dfrac{5!}{2!}=60$ (since L appears twice)

5️⃣ Takeaway Tips ✨

• Use factorials ($n!$) as the building block.
• When order doesn't matter, divide by the factorial of the number of items chosen to remove duplicate arrangements.
• For selections with unlimited supply, think of “stars and bars” – a classic combinatorial trick.
• Always double‑check whether items are distinct or identical; it changes the formula.