Fundamental principle of counting

Correct answer: D. 8 ways

Each of the 4 subjects can be paired with either of the 2 modes. By the multiplication principle, there are 4 × 2 = 8 ways.

Correct answer: A. 36

There are 9 choices for the drink. For each selected drink, there are 4 choices for the biscuit pack. Therefore, by the fundamental principle of counting, the total number of selections is \(9 \times 4=36\). Option 13 results from adding the numbers of items, but a drink and a pack form a pair, so multiplication—not addition—is required. Exam tip: Multiply the choices when each option in one step can be combined with every option in the next step.

Correct answer: B. 12 ways

For each of the 6 subjects, either of the 2 languages can be chosen. By the multiplication principle, the total is 6 × 2 = 12 ways.

Correct answer: C. 15 ways

Each of the 3 seal designs can be paired with any of the 5 signature styles. Thus, the multiplication principle gives 3 × 5 = 15 ways.

Correct answer: D. 14

For each of the 7 keyboard choices, there are 2 possible mouse choices. Therefore, by the fundamental principle of counting, the total number of selections is \(7 \times 2=14\). Adding them, \(7+2=9\), counts individual items, not keyboard-mouse pairs. Exam tip: Multiply the numbers of choices when one item is chosen from each separate group.

Correct answer: A. 15

There are 5 choices for the museum, and for each museum there are 3 choices for the lunch place. By the fundamental principle of counting, the total number of choices is \(5 \times 3=15\). Option 8, obtained from \(5+3\), only adds the available options; successive independent choices require multiplication. Exam tip: Multiply the numbers of choices when every first choice can be paired with every second choice.

Correct answer: B. 32 ways

Each of the 4 sizes can be combined with any of the 8 background colors. Therefore, there are 4 × 8 = 32 choices.

Correct answer: C. 12 ways

For each of the 2 identity-proof options, there are 6 address-proof options. Hence, the total number of pairs is 2 × 6 = 12.

Correct answer: D. 27

By the fundamental principle of counting, there are 9 choices for the group and 3 choices for the topic. Therefore, the total number of matches is \(9 \times 3=27\). The value 12 comes from \(9+3\), but independent choices are multiplied, not added. Exam tip: For successive choices connected by “and”, usually multiply the numbers of choices.

Correct answer: A. 24 watches

Each of the 6 straps can be paired with any of the 4 dials. By the multiplication principle, 6 × 4 = 24 different watches can be made.

Correct answer: B. 35

There are 5 choices for the subject category, and for each chosen category there are 7 choices for the marking scheme. By the fundamental principle of counting, the total number of choices is \(5 \times 7=35\). Option 49 comes from \(7 \times 7\), but there are only 5 categories. Exam tip: multiply the numbers of choices when independent selections are made in sequence.

Correct answer: C. (30) ways

This question uses the multiplication principle because each final identity-plate design contains two choices: one shape and one color. There are 3 possible shapes, and for each shape any of the 10 colors may be used. Therefore, the total number of shape-color combinations is 3 × 10 = 30. Option C is correct. The answer 13 comes from adding the numbers, which would count alternatives rather than complete designs. The value 20 has no basis in the stated choices, and 33 is another incorrect sum-like result. Both components are required, so their possibilities must be multiplied.

Correct answer: D. 24

There are 8 choices for the cover color and, for each chosen color, 3 choices for the page type. By the fundamental principle of counting, the total number of choices is \(8 \times 3=24\). Getting 11 would mean adding 8 and 3, but addition is used when choosing one option from separate alternatives. Exam tip: multiply the numbers of choices for successive independent selections.

Correct answer: A. 36 ways

The multiplication principle applies because each card design can be paired with any of the 9 envelope colors. Thus, the number of combinations is 4 × 9 = 36.

Correct answer: A. (40) codes

The direct answer is option A: 40 codes. The code has three ordered positions: the first position has 4 choices, the second has 5 choices, and the third has 2 choices. Since one choice is made at every position and the choices are independent, multiply them: \(4\times5\times2=40\). Option A is correct. Option B, 20, leaves out one factor and counts only \(5\times2\) or an equivalent incomplete calculation. Option C, 11, adds the numbers instead of counting combinations of choices; addition is not suitable when every position must be filled. Option D, 80, is twice the correct product and gives no valid interpretation of the stated choices. The order of positions matters because a letter choice in the first position is different from the same type of choice in another position. A reliable method is to draw three boxes and write 4, 5, and 2 above them, then multiply the numbers above the boxes.

Correct answer: B. 24 packs

The governing principle is the fundamental principle of counting: when a complete choice consists of independent stages, and the stages have m, n, and p possibilities, the total number of outcomes is m × n × p. Here there are 3 choices for the main dish, 4 choices for the drink, and 2 choices for the dessert. Therefore, total meal packs = 3 × 4 × 2 = 24. Each main dish can be paired with all 4 drinks and both desserts, so no combinations are omitted or repeated. Hence option B is correct. Adding the choices would not represent selecting one item from every category; 9, 12, and 18 arise from inappropriate operations.

Correct answer: C. 60 flags

The choices are made independently for the three stripe positions, so the fundamental counting principle gives 5 × 3 × 4 = 60 different flags.

Correct answer: D. (36) selections

The direct answer is option D: 36 selections. The student chooses one bag, one pen, and one notebook. These are three independent stages. There are 2 bag choices; for each bag there are 6 pen choices; for each bag-and-pen pair there are 3 notebook choices. Therefore the multiplication principle gives 2 × 6 × 3 = 12 × 3 = 36. Option D is correct. Option A, 11, is obtained by adding the numbers, but addition does not count complete selections containing one item from every category. Option B, 24, misses one factor or uses an incomplete calculation. Option C, 18, is also not the product of all three choice counts. The total is not the number of objects but the number of possible complete triples. Memory cue: for one choice from every independent group, multiply every group size once.

Correct answer: B. 40 plans

A plan includes one choice from each of the three categories. By the fundamental counting principle, the number of plans is 4 × 5 × 2 = 40.

Correct answer: A. (24) ways

The direct answer is option A: 24 ways. A complete uniform requires one shirt, one pair of trousers, and one pair of shoes. The choices are independent: 3 shirt choices, 2 trouser choices, and 4 shoe choices. For every shirt, there are 2 trouser choices, and for each such pair there are 4 shoe choices. Thus total complete uniforms = 3 × 2 × 4 = 6 × 4 = 24. Option A is correct. Option B, 9, comes from adding 3 + 2 + 4, which does not count combinations. Option C, 12, omits one factor or performs an incomplete product. Option D, 18, is not the product of the three available choices. No item is repeated and no restriction is stated, so every shirt can be paired with every trouser and every shoe. Exam cue: the word ‘complete’ means one selection from each required category, so use multiplication.

Correct answer: C. 84 passcodes

There are 7 choices for the first position, 4 for the second, and 3 for the third. Since one choice is made for each position, the total is 7 × 4 × 3 = 84 passcodes.

Correct answer: D. 60

To form one book choice, three independent selections are made: one of 6 subjects, one of 2 cover types, and one of 5 sizes. By the fundamental principle of counting, the total number of choices is \(6 \times 2 \times 5=60\). The value 30 comes from \(6 \times 5\) and incorrectly ignores the 2 cover-type choices. Exam tip: Multiply the number of choices when every choice at one stage can be combined with every choice at the next stage.

Correct answer: C. 72 ways

There are 4 choices for the class, 3 for the subject, and 6 for the test type. By the multiplication principle, the total is 4 × 3 × 6 = 72 ways.

Correct answer: A. 40

For each decoration, choose one of 5 wall colours, one of 2 curtain options, and one of 4 lamp options. Since these choices are independent, the multiplication principle gives \(5 \times 2 \times 4=40\) decorations. Adding them to get \(5+2+4=11\) is incorrect because choices are being made at successive stages. Exam tip: multiply the numbers of choices when independent selections occur in separate steps.

Correct answer: D. (60) ways

The governing concept is the fundamental principle of counting: when a task is completed through independent successive choices, the numbers of choices are multiplied. The kit requires one bat, one ball and one glove. There are 3 possible bat selections; for each selected bat, there are 5 possible ball selections, giving 3 × 5 = 15 bat-ball pairs. Each such pair can be combined with any of the 4 glove options, so the total is 3 × 5 × 4 = 60. Therefore option D is correct. Option A adds the numbers, while options B and C multiply only two of the three required choice stages and therefore do not count complete kits.