Fundamental principle of counting

The choices of coffee, cup size, and sugar level are independent. Therefore, by the fundamental principle of counting, the total number of choices is \(6 \times 3 \times 2=36\). Option 18 ignores the 2 possible sugar levels. Exam tip: Multiply the number of choices at each stage when every choice at one stage can be paired with every choice at the next stage.

This question uses the fundamental principle of counting. Choosing a room can be completed in three successive stages: first choose one of the 4 blocks, then one of the 5 floors in that block, and finally one of the 6 rooms on that floor. Since every block has the same number of floors and every floor has the same number of rooms, the choices multiply: 4 × 5 × 6 = 120. Therefore, option B is correct. Option A is only the product of 3 and 5, option C does not account for all three stages, and option D is an incorrect product. The order of making the choices does not change the total count.

By the fundamental principle of counting, each of the 3 classes can be combined with 4 departure times and 2 seat preferences. Hence, the total number of options is \(3 \times 4 \times 2=24\). The value 12 comes from \(3 \times 4\) and incorrectly ignores the seat preference. Exam tip: Multiply the numbers of choices when the selections are independent.

To form one message, choose 1 of the 5 greetings, 1 of the 3 main sentences, and 1 of the 4 closing sentences. By the fundamental principle of counting, the total number of messages is \(5 \times 3 \times 4=60\). An option such as 45 does not account for multiplication across all independent choices. Exam tip: multiply the numbers of choices when selections are made successively and independently.

The governing idea is the fundamental principle of counting. The coin has 2 possible outcomes, such as heads and tails, while the die has 6 possible outcomes, numbered 1 through 6. For each fixed coin result, the die can show any of its 6 results, so there are 6 pairs for that coin result. Repeating this for both coin results gives 2 × 6 = 12 ordered combined outcomes. Therefore, option A is correct. Adding 2 and 6 gives 8, but addition is not appropriate because one result from each experiment must be paired. Option C counts only die results, and option D incorrectly multiplies 6 by itself.

By the fundamental principle of counting, each of the 4 department codes can be paired with any of the 25 serial numbers. Hence, the total number of IDs is \(4 \times 25=100\). Getting \(29\) would mean adding the choices, but an ID requires both a code and a serial number, so the choices must be multiplied. Exam tip: multiply the numbers of choices when one choice is made at each successive stage.

For each head type, there are 4 choices of body type, and for every such choice, there are 5 choices of wheel type. Therefore, by the fundamental principle of counting, the total number of models is \(3 \times 4 \times 5=60\). The value 12 is only \(3 \times 4\) and ignores the wheel choices. Exam tip: Multiply the number of choices at each independent stage.

Each of the 2 letters can be paired with any one of the 10 digits. Therefore, by the fundamental principle of counting, the total number of house numbers is \(2 \times 10=20\). Adding the choices to get 12 is incorrect because one letter choice and one digit choice combine to form a pair. Exam tip: multiply the numbers of choices for successive independent selections.

To find the total number of seats, multiply the number of rows by the seats in each row: \(6 \times 7 = 42\). Therefore, the correct answer is 42. Option 13 comes from adding the two numbers, but equal groups are counted by multiplication. Exam tip: When equal groups are described using words such as “each” or “every”, multiplication is usually required.

For each practice set, choose one of 5 languages, one of 3 practice levels, and one of 2 recording modes. By the fundamental principle of counting, the total number of sets is \(5 \times 3 \times 2=30\). The value \(15\) counts only the language and level choices and ignores the recording mode. Exam tip: multiply the numbers of choices at independent stages.

One raga, one rhythm and one speed are chosen independently. Therefore, by the fundamental principle of counting, the total number of ways is \(4 \times 5 \times 2=40\). Adding them to get \(4+5+2=11\) is incorrect because one choice from each category is combined. Exam tip: multiply the numbers of choices when every stage is part of the same selection.

By the fundamental principle of counting, each of the 3 meal types can be paired with 6 discount options and, for each such pair, 4 validity-day options. Hence, the total number of coupon types is \(3 \times 6 \times 4=72\). Option 36 does not account for all 4 validity options. Exam tip: Multiply the numbers of choices when the selections are independent and made in stages.

This is a direct application of the fundamental principle of counting. To identify one computer, first select its row and then select its position within that row. There are 4 choices for the row. For every selected row, there are 9 choices for the computer, so the total number of row-position pairs is 4 × 9 = 36. Hence option A is correct. The answer 13 in option B comes from adding the two numbers, but addition would not count paired choices. Option C is only twice the number of rows, and option D is the sum of 4 and 9 multiplied by ? no valid counting rule; it is simply an incorrect distractor. Each computer corresponds to one unique row and position.

Since repetition is allowed, each of the three positions has 6 choices. By the fundamental principle of counting, the total number of codes is \(6\times6\times6=6^3=216\). The value 36 equals \(6^2\), which would apply to only two positions. Exam tip: multiply the choices for independent positions.

This uses the fundamental principle of counting because the selection is made in two successive stages. First, the main subject can be selected in 6 ways. After that choice, the same subject is unavailable, so the optional subject can be selected from the remaining 5 subjects in 5 ways. Since every main-subject choice can be paired with every permitted optional-subject choice, multiply the numbers: 6 × 5 = 30. Therefore, option C is correct. Adding 6 and 5 would count stages incorrectly, while 25 ignores the six possible main-subject choices. The value 36 would allow the same subject to be selected twice, which the condition explicitly forbids.

The governing idea is the multiplication principle with ordered choices. There are 5 possible choices for the first speaker. Once that student is selected, four students remain available for the second speaking position because repetition is not allowed. Therefore, the number of ordered selections is 5 × 4 = 20. The order matters: choosing student X first and student Y second is different from choosing Y first and X second. Thus option D is correct. The answer 10 would treat the pair as unordered, which is unsuitable for separate first and second positions. The value 25 incorrectly allows the first student to speak twice, and 15 does not follow the required two-stage count.

The governing concept is the fundamental principle of counting for independent choices. A completed form is determined by selecting one center, one time slot, and one language. The center has 4 choices, the time slot has 3 choices, and the language has 2 choices. Because each choice can be combined independently with every choice in the other categories, multiply them: 4 × 3 × 2 = 24. Therefore, option B is correct. Adding the numbers gives 9, but addition is used for alternatives rather than simultaneous selections. The value 12 omits the language choice, while 48 doubles the actual number by introducing an extra factor not stated in the problem.

This is a direct application of the multiplication principle. To identify one room, three successive choices are made: select one of the 3 blocks, then one of the 4 floors in that block, and finally one of the 8 rooms on that floor. Every block choice has 4 floor choices, and every floor choice has 8 room choices. Hence the total number of rooms, and therefore the number of ways to choose one room, is 3 × 4 × 8 = 12 × 8 = 96. Option C is correct. The value 64 ignores the blocks, 32 does not use the stated structure correctly, and 15 is merely an inappropriate sum of some quantities rather than a product of independent stages.

The governing concept is the multiplication principle for independent selections. One test setting is formed by choosing one subject, one difficulty level, and one language. There are 4 choices for the subject, 5 choices for the difficulty level, and 2 choices for the language. Since each subject can be paired with every difficulty level and either language, the total is 4 × 5 × 2 = 20 × 2 = 40 settings. Thus option B is correct. The value 20 omits the language choice, 11 incorrectly adds the three numbers, and 80 includes an extra factor not present in the question. The categories are chosen together, so multiplication rather than addition is required.

This problem uses the multiplication principle because two distinct roles must be filled. The captain must be a boy, so there are 6 choices for captain. The vice-captain must be a girl, so there are 4 choices for vice-captain. Every boy choice can be paired with every girl choice, giving 6 × 4 = 24 possible leadership pairs. Therefore, option D is correct. The result 10 comes from adding the group sizes and does not form pairs. The value 20 does not represent the required product, and 12 would also undercount the combinations. Since the roles and eligible groups are different, no further division is needed.

The governing concept is the multiplication principle for independent product features. A shirt type is completely specified by selecting one size, one color, and one print design. There are 4 possible sizes, 5 possible colors, and 3 possible print types. Each size can be combined with every color and every print, so the total number of distinct shirt types is 4 × 5 × 3 = 20 × 3 = 60. Therefore, option B is correct. The value 12 multiplies only size and print, leaving out color. The value 20 multiplies size and color but ignores print, while 45 does not arise from the stated independent choices. Because all three features are selected together, multiplication is required.

The governing concept is the fundamental principle of counting: when a choice is made in successive stages, and every choice at one stage can be followed by every choice at the next stage, the numbers of choices are multiplied. The student first has 5 independent chapter choices. After any one chapter is selected, there are 6 question choices within that chapter. Therefore, the total number of chapter-question pairs is 5 × 6 = 30. Option D is correct because each selection is identified by both its chapter and its question. Adding 5 and 6 would count stages rather than complete selections, while 25 and 36 do not represent the required product.

This is an application of the fundamental principle of counting. The first stage, tossing a standard coin, has 2 possible outcomes: head or tail. The second stage, rolling a standard die, has 6 possible outcomes, numbered 1 through 6. The final stage has 3 possible color choices. Every coin result can be paired with every die result and every color choice, so the total number of ordered experiment outcomes is 2 × 6 × 3 = 36. Thus option A is correct. The value 11 comes from addition, not multiplication; 18 omits one factor of 2, and 24 does not equal the product of all three independent stages.

The governing rule is the multiplication principle for independent successive choices. A complete code contains four positions, and a choice must be made at every position. The first position can be filled in 3 ways, the second in 4 ways, the third in 2 ways, and the fourth in 5 ways. Since each choice at one position may be combined with every permitted choice at the other positions, the number of complete codes is 3 × 4 × 2 × 5. Computing step by step gives 3 × 4 = 12 and 2 × 5 = 10, so the product is 12 × 10 = 120. Therefore option C is correct. Addition would count isolated choices rather than complete four-place codes.

The relevant concept is the fundamental principle of counting applied to the two digit positions. The tens digit has exactly 4 permitted choices: 2, 4, 6, and 8. For each selected tens digit, the units digit can independently be any one of the 5 odd digits: 1, 3, 5, 7, or 9. Thus every tens choice can be paired with every units choice, giving 4 × 5 = 20 valid two-digit numbers. Option D is therefore correct. No extra restriction is needed because all permitted tens digits are nonzero, so every constructed pair is automatically a two-digit number. The values 9, 16, and 25 fail to multiply the two independent sets correctly.