Fundamental principle of counting

Correct answer: B. 36 options

For each combination, one of 6 songs, one of 3 costumes, and one of 2 lighting effects is chosen. The multiplication principle gives 6 × 3 × 2 = 36 combinations.

Correct answer: A. 125 codes

Each of the 3 places has 5 choices, including when a choice is repeated. Thus, the number of codes is 5 × 5 × 5 = 125.

Correct answer: C. (400) plates

This uses the fundamental principle of counting, because a plate is formed by making four independent position-wise choices. The first letter has 4 choices and, since repetition is allowed, the second letter also has 4 choices. Similarly, the first digit has 5 choices and the second digit also has 5 choices. Thus the total number of plates is 4 × 4 × 5 × 5 = 16 × 25 = 400. Hence option C is correct. Repetition being allowed is important: it means that choosing a letter or digit does not reduce the choices at the next position. Option A, B and D result from incomplete or incorrect multiplication and do not represent all four positions.

Correct answer: D. (60) types

The governing idea is the fundamental principle of counting. A complete registration type is determined by three independent selections: one course, one batch and one payment option. There are 3 choices for the course. For every chosen course, any of the 4 batches may be selected, producing 3 × 4 = 12 course-batch combinations. Each of those combinations can be paired with any of the 5 payment options. Therefore the total is 3 × 4 × 5 = 60 registration types. Option D is correct. Option A counts only course-batch pairs, option B multiplies only two of the given numbers, and option C is an incorrect product; none of these counts includes all three required decisions correctly.

Correct answer: B. 30

By the fundamental principle of counting, each fruit choice can be combined with 5 chocolate choices, and each such combination has 3 packing choices. Therefore, the total number of ways is \(2 \times 5 \times 3=30\). Option 15 ignores the packing choices. Exam tip: multiply the numbers of choices when one option is selected at each independent stage.

Correct answer: A. 24 routes

Each of the 4 routes to the market can be followed by any of the 6 routes to school. Therefore, there are 4 × 6 = 24 route combinations.

Correct answer: C. 28 options

Each of the 7 chapter choices can be paired with any of the 4 question types. Thus, there are 7 × 4 = 28 practice options.

Correct answer: B. (243) ways

The direct answer is option B: 243 ways. One question must be selected from each of 5 sections. Each section offers 3 choices, so the process has five stages, each with 3 possibilities. By the multiplication principle, the total is 3 × 3 × 3 × 3 × 3 = 3^5 = 243. Option B is correct. Option A, 15, adds 3 five times; that counts neither the possible complete selections nor the combinations of choices. Option C, 125, is 5^3, which reverses the base and exponent and does not represent five stages with three choices each. Option D, 81, equals 3^4, so it counts only four sections, not five. The phrase ‘from each section’ means every section must contribute one answer. Memory cue: repeated equal choices are written as a power: choices per stage raised to number of stages.

Correct answer: D. (16) codes

The relevant concept is the fundamental principle of counting. A four-place code is formed by filling four positions one after another. Each position has 2 choices, and repetition is allowed, so the number of choices remains 2 at every position. Hence the total number of codes is 2 × 2 × 2 × 2, which can be written as 2^4 = 16. Therefore option D is correct. The fact that repetition is allowed prevents the choices from decreasing after an earlier selection. Option A adds the choices, option B gives only 2^3, and option C does not follow the multiplication rule. Every valid code must account for all four positions, so 16 is the complete count.

Correct answer: C. 60

For each language option, there are 5 time-limit options, and for every such choice there are 3 mark options. Therefore, by the fundamental principle of counting, the total number of combinations is \(4 \times 5 \times 3=60\). Option 20 comes from \(4 \times 5\) only and ignores the mark options. Exam tip: multiply the number of choices at every independent step.

Correct answer: A. (24) forms

The direct answer is option A: 24 forms. A painting is determined by three separate selections: one background color, one border color, and one signature color. There are 3 choices for the background, 4 for the border, and 2 for the signature. Therefore, by the multiplication principle, the total number is \(3\times4\times2=24\). Option A is correct because each background can be combined with every border and every signature color. Option B, 9, comes from an unsuitable addition and does not count all combinations. Option C, 12, omits one factor or counts only part of the choices. Option D, 48, is twice the correct total and would require an extra independent choice that the question does not provide. The colors are treated as choices for different roles; even if the same color name could appear in more than one role, each role is still selected from its stated list. The memory rule is simple: separate required selections made together are multiplied.

Correct answer: B. (60) designs

The governing concept is the fundamental principle of counting: independent design choices are multiplied to obtain the number of complete designs. A certificate can use 1 of 6 title styles, 1 of 2 seal options and 1 of 5 border designs. For each title style there are 2 possible seals, giving 6 × 2 = 12 title-seal combinations. Each of these can be paired with any of the 5 borders, so the total is 6 × 2 × 5 = 60. Thus option B is correct. Option A adds the choices instead of multiplying, option C omits the factor of 2, and option D doubles the correct product without justification. All three independent features must be included.

Correct answer: D. 32 ways

There are 8 choices for the shelf and 4 choices for a row on the selected shelf. By the multiplication principle, 8 × 4 = 32 ways.

Correct answer: A. 30

For each seat category, there are 5 choices of bus time and 2 choices of payment mode. By the fundamental principle of counting, the total number of options is \(3 \times 5 \times 2=30\). The value \(10\) counts only \(5 \times 2\) and ignores the seat categories. Exam tip: When choices are made in independent stages, multiply the number of choices at each stage.

Correct answer: C. 63

There are 9 choices for the first place. For each chosen colour, any one of 7 shapes can be used in the second place. Therefore, by the fundamental principle of counting, the total number of signals is \(9 \times 7=63\). Option 16 comes from adding \(9+7\), but independent choices made in successive places are multiplied, not added. Exam tip: Multiply the numbers of choices when a result is formed through consecutive selection steps.

Correct answer: D. 24

There are 4 choices for captain. For each such choice, there are 3 choices for vice-captain and 2 choices for manager. By the fundamental principle of counting, the total number of arrangements is \(4 \times 3 \times 2=24\). Adding the choices, \(4+3+2=9\), is incorrect because all three posts are being filled together. Exam tip: multiply the numbers of choices when successive selections are made.

Correct answer: B. (60) types

The governing concept is the fundamental principle of counting. Each complete T-shirt type is determined by making one choice from each of three independent categories: color, size, and print. There are 5 choices for color. For every selected color, there are 4 possible sizes, giving 5 × 4 = 20 color-size combinations. Each of these combinations can be paired with any of the 3 prints. Therefore, the total number of distinct T-shirt types is 5 × 4 × 3 = 60, so option B is correct. Option A adds the category counts, option C omits the print choices, and option D is an incorrect product. Since all three features are required, all three factors must be multiplied.

Correct answer: A. (30) settings

The governing principle is multiplication for independent choices. To create one setting, the teacher must select both a chapter and a difficulty level. There are 6 possible chapter choices. For every selected chapter, any of the 5 difficulty levels can be used, so each chapter produces 5 settings. Consequently, the total is 6 × 5 = 30 settings. Option A is correct. Adding 6 and 5 gives option B, but addition would count alternatives rather than a combined chapter-level pair. Option C is the square of 5 and ignores the six chapters, while option D is an unsupported product. Because both choices are required simultaneously, multiplication is the appropriate operation.

Correct answer: C. 48

To form one poster style, choose one of 4 fonts, one of 6 colours, and one of 2 background textures. By the fundamental principle of counting, the total number of styles is \(4 \times 6 \times 2=48\). Option 24 misses the multiplication by the 2 texture choices. Exam tip: Multiply the numbers of choices when every choice at one stage can be combined with every choice at the next stage.

Correct answer: B. 42

For each pass type, there are 7 colour choices, and for each colour, there are 3 validity-period choices. Therefore, by the fundamental principle of counting, the total number of passes is \(2 \times 7 \times 3=42\). Adding \(2+7+3=12\) is incorrect because these options are selected together, not as alternatives. Exam tip: Multiply the numbers of choices when independent selections are made in sequence.

Correct answer: D. 40 ways

For each of the 5 rows, there are 8 seats to choose from. By the fundamental principle of counting, the total number of choices is 5 × 8 = 40.

Correct answer: A. (60) looks

The fundamental principle of counting applies because each final look is created by making one independent choice from each of three groups. There are 3 filter choices, 4 frame choices and 5 sticker choices. First, a filter and frame can be paired in 3 × 4 = 12 ways. Every one of these 12 pairs can be combined with any of the 5 stickers, giving 12 × 5 = 60 looks. Equivalently, the count is 3 × 4 × 5 = 60. Therefore option A is correct. Option B stops after two features, option C uses only 3 × ? Actually 20 omits one factor, and option D is an incorrect product. All three selections are required for a complete look.

Correct answer: B. 36

For each of the 4 optional subjects, any one of the 9 project topics can be chosen. Therefore, by the fundamental principle of counting, the total number of selections is \(4 \times 9=36\). Adding, \(4+9=13\), only gives the total number of available items, not the number of subject–project pairs. Exam tip: When one choice is made at each of two independent stages, multiply the numbers of choices.

Correct answer: C. (60) setups

The governing concept is the fundamental principle of counting. A complete test setup requires one mode, one question-count choice and one time choice. The mode can be selected in 2 ways. For each mode, there are 5 question-count options, giving 2 × 5 = 10 mode-count combinations. Each combination can be paired with any of the 6 time options. Thus the total number of setups is 2 × 5 × 6 = 60, making option C correct. Option A adds the numbers rather than forming combinations. Option B omits the time factor, and option D is an incorrect product. Since every setup must specify all three independent settings, multiplication is required.

Correct answer: D. (144) PINs

This uses the fundamental principle of counting with different numbers of choices at different positions. The first position has 3 choices and the second position also has 3 choices, so the first two positions contribute 3 × 3 = 9 possibilities. The third position has 4 choices and the fourth also has 4 choices, contributing 4 × 4 = 16 possibilities. Because repetition is allowed, these choices remain available independently. Therefore the total number of PINs is 3 × 3 × 4 × 4 = 9 × 16 = 144. Option D is correct. Options A, B and C result from omitting one factor or using an incorrect combination, so they do not count all four positions.