2 Counting Techniques Free Online FlashCards

What does the addition principle count?

For mutually exclusive alternatives, add the numbers of choices: $m+n$.

How do you count overlapping categories?

Subtract the overlap: $|A\cup B|=|A|+|B|-|A\cap B|$.

What does the multiplication principle count?

Multiply the numbers of choices at successive stages: $n_1n_2\cdots n_k$.

What is a factorial?

For a positive integer, $n!=n(n-1)\cdots2\cdot1$; by definition, $0!=1$.

What is a permutation?

A permutation is an arrangement in which order matters. The count is $P(n,r)=\frac{n!}{(n-r)!}$.

How are arrangements with repeated objects counted?

For repeated objects, divide by the factorial of each repetition count: $\frac{n!}{n_1!n_2!\cdots n_k!}$.

What is a combination?

A combination is an unordered selection. Its count is $\binom{n}{r}=\frac{n!}{r!(n-r)!}$.

How do you choose between permutations and combinations?

Use a permutation when changing order creates a different outcome; use a combination when it does not.

How are permutations and combinations related?

The relationship is $P(n,r)=\binom{n}{r}r!$: first choose the objects, then arrange them.

What is the finite equally likely probability formula?

For equally likely outcomes, $P(A)=\frac{|A|}{|S|}$, where $|A|$ is favorable outcomes and $|S|$ is total outcomes.

How many five-card hands are possible?

The number of five-card hands is $\binom{52}{5}=2{,}598{,}960$, because deal order does not matter.

How is exactly two aces counted in a five-card hand?

Count favorable hands as $\binom{4}{2}\binom{48}{3}$, then divide by $\binom{52}{5}$.