Fundamental principle of counting

The two digit positions have (10) and (10) choices, and the letter has (5) choices. Total ways are (10\times10\times5=500).

Each position has (6) choices, so (6^4=1296). When the same choices repeat, using powers is faster.

The four positions have (4,3,2,1) choices, so (4\times3\times2\times1=24). If all digits are used, choices decrease step by step.

For an even number, the units digit can only be 0 or 2. If the units digit is 0, there are 3 choices (1, 2, 3) for the hundreds digit and 2 choices for the tens digit: \(3\times2=6\). If the units digit is 2, 0 cannot be in the hundreds place, so there are 2 choices (1, 3) for it and 2 choices for the tens digit: \(2\times2=4\). Hence, the total is \(6+4=10\). Option 8 results from an incorrect treatment of 0 in the hundreds place. Exam tip: the first digit of a multi-digit number cannot be 0.

The governing concept is the addition rule of counting, which applies when exactly one choice is made from mutually exclusive groups. A customer chooses either one tea, one coffee, or one juice; the choices are alternatives rather than simultaneous selections. Therefore, the total number of possible drinks is 3 + 4 + 2 = 9. Option C is correct because it counts every available drink exactly once. Multiplication would be appropriate only if the customer had to choose one item from each category, which would give 3 × 4 × 2 = 24, but that is not what the question asks. Thus, options A and B use an inappropriate multiplication or partial count, while option D omits two choices.

Both color and size are to be selected, so (5\times4=20). Multiply when product features are independent.

The three stages are independent, so (3\times2\times4=24). In design questions, treat each feature as one stage.

The student must choose only one course, either online or offline. These are mutually exclusive choices, so apply the addition rule: \(6+3=9\). Hence, there are 9 possible choices. \(18\) would result if one online course and one offline course had to be chosen together; that would use the multiplication rule. Exam tip: For “or” with exactly one choice, usually add the numbers of alternatives.

One picture and one message are both needed, so (4\times5=20). The multiplication principle applies to combined selection.

There are 3 choices of colour, and for each colour there are 4 choices of capacity. Therefore, by the multiplication principle, the total number of models is \(3\times4=12\). Adding the choices to get 7 is incorrect because every colour can be paired with every capacity. Exam tip: multiply the numbers of choices when independent selections combine to form one outcome.

The governing concept is the fundamental principle of counting. If a procedure has successive independent stages, and every choice at one stage can be paired with every choice at the next stage, the total number of outcomes is found by multiplication. There are 4 possible choices for monitor. For each chosen monitor, there are 3 possible choices for assistant monitor. Hence the total is 4 × 3 = 12. The positions are distinct, so choosing candidate X as monitor and candidate Y as assistant monitor is a role assignment, not an unordered selection of five people. Option B, 7, incorrectly adds the choices. Options C and D do not follow the multiplication rule. Therefore option A is correct.

There are (5) choices first and (2) choices next, so (5\times2=10). Multiply the stages in a sequential journey.

The direct answer is C: 18 profiles. A profile requires one photo choice and one username-style choice. For every one of the 3 photo options, any of the 6 username styles can be selected. Therefore the independent choices are multiplied: \(3\times6=18\). Option A, 9, incorrectly adds 3 and 6; addition is for alternatives where only one group is chosen. Option B, 12, has no valid counting rule here and undercounts the combinations. Option C is correct because it includes every photo-style pairing. Option D, 36, is too large and would count extra choices not present in the question. The reliable method is to list the stages: first choose a photo, then choose a username style, and multiply the numbers of possibilities.

This uses the multiplication rule, also called the fundamental principle of counting. A complete avatar requires one face, one hat, and one costume, so a choice in one category is combined with a choice in each of the other independent categories. The total is 2 × 4 × 3 = 24. Hence option D is correct. The number 12 would count only two groups, such as faces and hats, and would ignore the costume choices. The number 9 is an addition-rule result, which would be suitable only if the user selected exactly one item from one of the categories rather than building a complete avatar. The value 18 has no valid counting basis here. Since every face can be paired with every hat and every costume, multiplication counts all distinct combinations without omission.

This uses the fundamental principle of counting. The choice is made in two successive stages: first select one shirt, which can be done in 3 ways, and then select one trouser, which can be done in 4 ways. Each shirt can be paired with each trouser, so the total number of complete outfits is 3 × 4 = 12. Therefore, option B is correct. Addition, 3 + 4 = 7, would count alternatives when only one item is selected, not a shirt-and-trouser pair. The value 9 has no applicable counting basis, and 24 would require more choices than the two given stages.

Each of the 5 tea choices can be paired with any of the 3 snack choices. By the multiplication principle, the total is 5 × 3 = 15 ways.

There are 4 choices for the first position. For each of these choices, there are 6 choices for the second position; hence, by the fundamental principle of counting, the total number of codes is \(4\times6=24\). Option 10 comes from adding the choices, which is incorrect because every letter can be paired with every digit. Exam tip: multiply the numbers of choices for successive independent selections.

The governing concept is the fundamental principle of counting, also called the multiplication rule. A complete outcome requires two selections: one bus route and one train route. There are 2 choices for the bus route, and for each of those choices there are 5 independent choices for the train route. Thus the number of ordered bus-train pairs is 2 × 5 = 10. Therefore option D is correct. Adding 2 and 5 to obtain 7 would be appropriate only if the person chose one item from either category, not one from both. The answer 5 ignores the bus selection, while 20 has no basis in the stated two-stage process. Each pair represents one complete travel choice.

Each of the 6 cap-color choices can be paired with either of the 2 ribbon-color choices. Thus, 6 × 2 = 12 ways.

There are 4 choices for the vegetable dish, and for each such choice there are 3 choices for the bread. By the fundamental principle of counting, the total number of selections is \(4 \times 3=12\). The value 7, obtained by \(4+3\), counts the available items, not the possible pairs. Exam tip: When one item is chosen from each separate category, multiply the numbers of choices.

Each of the 3 paper choices can be paired with any of the 5 ribbon choices. By the multiplication principle, there are 3 × 5 = 15 ways.

The direct answer is option C: 14 combinations. The student must make two successive choices: first choose one pen, then choose one notebook. There are 7 choices for the pen and, for each chosen pen, 2 choices for the notebook. By the fundamental principle of counting, multiply independent choices: 7 × 2 = 14. Option C is correct. Option A, 9, comes from adding 7 and 2; addition is not suitable when one item from each group is selected. Option B, 12, is an incorrect product. Option D, 16, is also not the product of the available choices. Each pen can pair with each notebook, giving pairs such as pen 1 with notebook 1 or 2, and so on: 7 groups of 2 pairs. Memory cue: ‘and’ usually means multiply choices; ‘or’ often means add mutually exclusive choices.

The fundamental principle of counting applies because a valid selection contains two components: one cover and one screen guard. The cover can be selected in 4 ways. For each one of those covers, any of the 4 screen guards may be selected, giving 4 × 4 = 16 possible pairs. Thus option D is correct. Adding 4 + 4 = 8 would describe a choice between a cover and a guard rather than choosing both. The answer 4 counts only one category, and 12 does not represent the number of combinations formed by the two independent stages. Equal category sizes do not change the multiplication rule.

For each of the 2 jackets, any of the 6 pairs of shoes can be chosen. By the fundamental counting principle, the total is 2 × 6 = 12 outfits.

There are 3 choices for the first section and 7 for the second. By the fundamental counting principle, the number of pairs is 3 × 7 = 21.