Agenda

<aside> 📌 Reminders

• Last assignment! Due December 1, 2024
• Last tutorial
• Final exam December 6, 2024 </aside>

Permutations of Distinct Objects

Recall from last week,

• A permutation is an arrangement of objects in a specific order.
• The number of permutations of $n$ distinct objects is $n! = n \cdot (n-1) \cdot (n-2) \cdot \dots \cdot 1$
• $r$-Permutations: If we are arranging $r$ objects chosen from $n$, the formula is: $P(n, r) = \frac{n!}{(n-r)!}$

Combinations of Distinct Objects

• A combination is a selection of objects without regard to order.
• The number of ways to choose $r$ objects from $n$ distinct objects is:

$$ C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}  $$

• Key point: Combinations differ from permutations because order does not matter.

Permutations with Indistinguishable Objects

• If some objects in a set are indistinguishable, the number of permutations is reduced.
• For $n$ objects, where there are $n_1, n_2, \dots, n_k$ indistinguishable objects of $k$ different types:

$$ \text{Number of permutations} = \frac{n!}{n_1! \cdot n_2! \cdot \dots \cdot n_k!}  $$

• Example: Arranging the letters in "BALLOON":
  • Total letters: $n = 7$
  • Repeated letters: $L = 2$, $O = 2$
  • Permutations: $\frac{7!}{2! \cdot 2! \cdot 1! \cdot 1! \cdot 1!} = 1260$

Combinations with Indistinguishable Objects