Lesson 33: Applying Counting Principles

New Concept

If there are $n_1$ ways to choose the first item, $n_2$ ways to choose the second item, and so on, then there are $n_1 \cdot n_2 \cdot \ldots \cdot n_k$ ways to choose all $k$ items.

What’s next

Next, you'll apply this principle to calculate possibilities for passwords, sports lineups, and other combinations.

Suppose a trial can result in any of $n_1$ outcomes from one category, any of $n_2$ outcomes from another category, and so on. If there are $k$ different categories of outcomes, then the total number of outcomes that can result is $n_1 + n_2 + \ldots + n_k$.

Examples

• Draw a king or a queen from a standard deck: 4 kings + 4 queens = 8 possible outcomes.
• From numbers 1-20, choose a perfect square or a multiple of 7: \{1, 4, 9, 16\} or \{7, 14\}. This gives $4 + 2 = 6$ total outcomes.

Suppose $k$ items are to be chosen. If there are $n_1$ ways to choose the first item, $n_2$ ways to choose the second item, and so on, then there are $n_1 \cdot n_2 \cdot \ldots \cdot n_k$ ways to choose all $k$ items.

Examples

• Create a 2-character password using letters (A-Z) where letters can be repeated: $26 \cdot 26 = 676$ possible passwords.
• An ice cream shop has 10 flavors and 4 types of toppings. The number of single-scoop, single-topping options is $10 \cdot 4 = 40$.
• A student has 5 shirts and 3 pairs of pants. The number of different outfits is $5 \cdot 3 = 15$.

Two events are independent if the probability of one event is not affected by whether or not the other event occurs. Two events are dependent if the probability of one event is affected by whether or not the other event occurs.

Examples

• Independent: Flipping a coin and then rolling a die. The coin's outcome ($H$ or $T$) has no effect on the die's outcome (1-6).
• Dependent: Drawing a card from a deck, not replacing it, and then drawing another. The first card drawn changes the remaining deck.
• Independent: Picking a marble from a bag, putting it back, then picking another. The odds reset for the second pick.