Combinatorics and probabilities

When a choice is made in two independent steps, use the multiplication principle.

1. Count the possibilities for the first choice.
2. For each of them, count the possibilities for the second choice.
3. The total number of possibilities is their product: m · n.

Add only when choosing one of separate alternatives. If every first choice can be paired with every second choice, multiply the numbers of choices.

Choose either one type of butter or one spread. These choices are alternatives rather than pairs, so add their counts. For example, 2 types of butter or 3 spreads give 2 + 3 = 5 choices.

The equally likely outcomes are sectors, not distinct labels. Count every sector with the target number and divide by the total number of sectors.

Split the sandwiches into two separate groups.

1. For a sandwich with butter and cheese, each type of butter can be paired with each type of cheese. Therefore, multiply the number of butter types by the number of cheese types.
2. A sausage-only sandwich has no butter or cheese choice, so the number of these sandwiches equals the number of sausage types.
3. Add the results of the two groups.

Rule: butter types × cheese types + sausage types.

Split the sandwiches into two separate groups: butter with cheese and butter with sausage.

1. In the first group, each butter type can be paired with each cheese type, giving butter types × cheese types combinations.
2. In the second group, each butter type can be paired with each sausage type, giving butter types × sausage types combinations.
3. Add the results for the two separate groups.

Rule: butter types × cheese types + butter types × sausage types.

Count the cards whose numbers are divisible by the stated divisor. Divide by the total number of cards and simplify the fraction.

Each kebab contains one choice of meat, one choice of salad and one choice of sauce.

1. Each type of meat can be paired with every salad choice.
2. Each resulting pair can be combined with every sauce.
3. Therefore, multiply the numbers of all three choices.

Rule: number of meat types × number of salad choices × number of sauces.