Fundamental principle of counting

The governing concept is the multiplication principle of counting. The student makes two successive choices: one shirt and one trouser. For each of the 4 shirt choices, any of the 3 trouser choices can be paired with it. Hence the total number of complete outfits is 4 × 3 = 12. Option C is correct. Adding 4 and 3 gives 7, but addition is used for alternatives such as choosing a shirt or a trouser, not for making both choices together. The value 9 has no counting basis here, and 16 would incorrectly pair four choices with four choices rather than three. Since the question uses “and” and asks for a combined selection, multiplication is the appropriate rule.

The first place has (5) choices and the second has (10), so (5\times10=50). Count choices for each position separately.

The choice is to select exactly one speaker from two non-overlapping groups: 7 boys or 5 girls. When one item is chosen from alternatives, the addition principle applies. There are 7 possible choices if the speaker is a boy and 5 possible choices if the speaker is a girl. Since these cases cannot occur simultaneously for one selected person, the total number of choices is 7+5=12. Therefore option C is correct. Multiplying 7×5=35 would count paired choices of one boy and one girl, which is a different problem involving two selections. The value 2 merely counts the groups and 24 does not follow from the stated choice.

The governing concept is the addition principle of counting. The selected book must be either one of the 8 mathematics books or one of the 6 science books. These are mutually exclusive categories for a single choice, so their counts are added: 8 + 6 = 14. Therefore option B is correct. Multiplying 8 by 6 would count pairs consisting of one mathematics book and one science book, but the question asks for only one book. Choosing 8 or 6 alone ignores the other category and gives an incomplete total. Because no book is counted in both categories, there is no overlap to subtract. The wording “one book” and the alternative “mathematics or science” signal that addition, not multiplication, is required.

The governing concept is the multiplication principle for consecutive stages. Every route from A to C consists of one choice for travelling from A to B followed by one choice for travelling from B to C. For each of the 3 first-stage routes, there are 4 possible second-stage routes. Thus the total is 3 × 4 = 12, so option D is correct. Adding 3 and 4 would treat the two stages as alternatives, but both stages are required in the same journey. The value 16 uses the wrong number of choices, and 9 has no corresponding product or valid counting interpretation. The routes are counted as ordered pairs: first-stage route, second-stage route. Assuming the route choices are available independently, the multiplication rule gives the complete count without omission or repetition.

The governing concept is the multiplication principle for a complete arrangement with several independent choices. A complete gift pack requires one cover, one ribbon, and one card. There are 5 choices for the cover; for each of those, 2 ribbon choices remain; and for each resulting pair, 3 card choices are possible. Therefore the total number of packs is 5 × 2 × 3 = 30. Option D is correct. The value 10 counts only covers and ribbons, leaving cards unselected. The value 20 does not represent all three stages, and 25 incorrectly treats the choices as though two categories each had five options. Since every pack includes one item from each category, the choices are consecutive and must be multiplied rather than added.

Each position has (10) choices, so (10\times10\times10=1000). In a code, if leading (0) is allowed, take (10) choices.

The direct answer is option B, 81. A two-digit passcode has two positions: the first position and the second position. Digits 1 through 9 give 9 choices for the first position. Repetition is allowed, so after choosing the first digit, all 9 choices are still available for the second position. By the fundamental principle of counting, multiply the independent choices: 9×9=81. Therefore, 81 two-digit passcodes can be formed. Option A, 18, is wrong because it adds the choices instead of multiplying them, and it does not represent two successive positions. Option B, 81, is correct. Option C, 72, would be 9×8, which would apply if repetition were not allowed; that is not the condition here. Option D, 90, does not follow from the available choices and is too large. Notice that this is a passcode, not necessarily a two-digit number selected without a leading-zero issue; in any case, the allowed digits are 1 to 9, so each position has exactly 9 choices. Examples include 11, 12, 19, 91, and 99, all allowed because repetition is permitted. Memory cue: “and” or successive slots usually means multiply; “repetition allowed” means the number of choices stays the same in each slot.

The first place has (4) choices and the second has (3), so (4\times3=12). Without repetition, choices decrease at the next place.

The tens place cannot be (0), so it has (4) choices and the units place has (5). Total ways are (4\times5=20).

The hundreds place cannot be (0), so it has (3) choices. Then (3) and (2) choices remain, giving (3\times3\times2=18).

Each place has (4) choices, so (4\times4\times4=64). With repetition, each place keeps the same choices.

The units place has (3) choices from (2,4,6), and (5) choices remain for the tens place. Total ways are (3\times5=15).

The units place has (3) choices (1,3,5), and the tens place has (5) choices from (1) to (5). Total ways are (5\times3=15).

Each position can take (5) letters, so (5\times5=25). If repetition is allowed in a code, keep the same choices for each position.

The governing concept is the multiplication principle applied to fixed code positions. The code has four positions: the first, second, and third letters each have 5 choices, and the final digit has 8 choices. Because repetition is allowed, each letter position independently retains all 5 choices. Hence the number of codes is 5 × 5 × 5 × 8 = 5³ × 8 = 125 × 8 = 1000. Option D is correct. The value 125 counts only the three-letter portion and omits the digit. The value 40 might arise from multiplying only two categories, but it does not account for the three separate letter positions. The value 625 uses four positions with five choices and ignores that the final position has eight choices. Independence and repetition make multiplication valid for every position.

The direct answer is option A, 12. The experiment has two parts performed together: tossing a coin and rolling a die. A coin has 2 possible outcomes, usually heads or tails. A standard die has 6 possible outcomes, numbered 1 through 6. For every coin result, each of the 6 die results can occur. Therefore, by the multiplication principle, total outcomes=2×6=12. They can be listed as H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6. Option A, 12, is correct. Option B, 8, is wrong because it does not result from combining all 2 coin outcomes with all 6 die outcomes. Option C, 6, counts only the die outcomes and ignores the coin. Option D, 2, counts only the coin outcomes and ignores the die. This is a case of “and”: one coin outcome and one die outcome together form a complete outcome, so multiply rather than add. Addition would be used for separate alternatives such as coin or die, not for both experiments occurring in one trial. The outcomes are equally structured as ordered pairs, such as (H,4) or (T,2). Memory cue: when independent experiments happen together, multiply their numbers of outcomes.

Each coin has two possible outcomes: head or tail. Since the coins are distinct, each outcome of the first coin can be paired with 2 outcomes of the second coin. Hence, the total number of outcomes is \(2\times2=4\): (H, H), (H, T), (T, H), and (T, T). Option 2 counts the outcomes of only one coin. Exam tip: Multiply the numbers of independent choices for distinct objects.

The first die has (6) outcomes and the second has (6), so (6\times6=36). In ordered outcomes, the roles of both dice are distinct.

For each of the 4 colors, any one of the 3 shapes can be chosen. Therefore, by the fundamental principle of counting, the total number of designs is \(4\times 3=12\). Option 7 results from adding the numbers of colors and shapes, but a design requires one choice of each, so the numbers must be multiplied. Exam tip: For successive independent choices, multiply the number of available choices at each step.

The first stage has (3) choices and the second has (4), so (3\times4=12). Fill password positions in order.

The direct answer is B: 11 choices. The student must choose exactly one topic, so the alternatives are poetry or painting, not both together. There are 5 poetry choices and 6 painting choices. By the addition principle, separate alternatives are added: \(5+6=11\). Option A, 30, comes from \(5\times6\), which would be appropriate if one poetry topic and one painting topic had to be chosen together. Option B is correct because only one choice is made from the two separate groups. Option C counts only the painting topics and ignores poetry. Option D counts only poetry and ignores painting. The cue is simple: use addition for either-or choices and multiplication for and-together choices.

The direct answer is option C, 7. The trip requires choosing exactly one place: either one of 3 museums or one of 4 parks. These are separate alternatives, so the addition principle applies. Add the numbers of choices: 3+4=7. Therefore, there are 7 total options. Option A, 12, is wrong because multiplication would describe choosing one museum and one park together, but the wording says one place, not both. Option B, 9, is simply not the correct sum and does not follow from the given groups. Option C, 7, is correct because the 3 museum choices and 4 park choices are mutually exclusive alternatives. Option D, 1, would mean there is only one possible destination, which ignores all the listed choices. The important word is “or.” When we choose one item from one group or one item from another separate group, add the choices. For example, if museums are M1, M2, M3 and parks are P1, P2, P3, P4, the complete list has seven possibilities: M1, M2, M3, P1, P2, P3, P4. We do not multiply because the trip does not select a museum and a park at the same time. Memory cue: “or” for separate choices usually means addition; “and” for successive choices usually means multiplication.

For each language, there are 2 possible theme choices. Therefore, for 4 language options, the total number of selections is \(4\times2=8\). Adding to get \(6\) is incorrect because one language and one theme form a pair, so the multiplication principle applies. Exam tip: Multiply the numbers of choices when independent selections are made in sequence.

There are 2 choices for the bag. For each bag, there are 5 choices for the water bottle, and for every such pair, there are 3 choices for the lunch box. Therefore, by the fundamental principle of counting, the total number of ways is \(2\times5\times3=30\). The value 10 is only \(2\times5\) and ignores the lunch-box choices. Exam tip: When one item is chosen from each independent category, multiply the numbers of choices.