Fundamental principle of counting

Apply the multiplication principle because the required outcome contains one choice from each of two categories. There are 5 possible bottle colors. After selecting any one bottle color, there are 2 possible box sizes, so every bottle choice can be paired with both box choices. The total number of bottle-box pairs is therefore 5 × 2 = 10, making option C correct. The value 7 comes from adding 5 and 2, which would count alternatives rather than complete selections. The value 25 incorrectly squares the first number, and 12 is not produced by the stated independent choices. Since the wording requires both one bottle and one box, multiplication is the appropriate operation.

Each of the 8 choices for the first position can be combined with any of the 3 choices for the second. Thus, 8 × 3 = 24 passwords are possible.

Each of the 4 cities can be paired with any of the 5 hotels. Therefore, the number of plans is 4 × 5 = 20.

The direct answer is B: 18 pairs. A program pair requires two independent choices: one song and one dance. There are 6 choices for the song. For each chosen song, any of the 3 dances may be selected. By the fundamental principle of counting, multiply the choices: \(6\times3=18\). Option A, 9, adds the choices and does not count all combinations. Option B, 18, is correct because every song can be paired with every dance. Option C, 12, has no correct counting basis here. Option D, 6, counts only the song choices and ignores the dance choice. For example, song 1 can form three pairs, and the same is true for all six songs, giving \(3+3+3+3+3+3=18\). Memory cue: use addition for either/or choices, but multiplication when choices are made together.

Each of the 2 backgrounds can be paired with any of the 9 frames. By the fundamental counting principle, there are 2 × 9 = 18 combinations.

Making one badge involves two independent choices: 1 of 5 shapes and 1 of 4 colors. Each shape can be paired with any of the 4 colors, so by the fundamental principle of counting, the total number of choices is \(5 \times 4 = 20\). Adding to get 9 is incorrect because independent successive choices are multiplied. Exam tip: For independent steps joined by “and”, usually multiply the numbers of choices.

This is a direct application of the fundamental principle of counting. The student must make both choices: one subject and one time slot. There are 3 possibilities for the subject, and for each selected subject there are 8 possible time slots. Hence each subject can be paired with every slot, giving 3 × 8 = 24 subject-slot combinations. Therefore option A is correct. The number 11 results from adding the category sizes, but addition would describe a choice between categories rather than a choice from both. The values 16 and 32 are not supported by the two independent stages. The product counts every valid complete schedule exactly once, so no further adjustment is needed.

For each of the 7 novels, any of the 5 magazines can be chosen. Thus, the number of novel-and-magazine pairs is 7 × 5 = 35.

The relevant concept is the multiplication principle of counting. One complete identity-card design requires one color-scheme choice and one font-style choice. There are 4 possible color schemes, and each of them may be combined with any of the 6 font styles. Consequently, the number of distinct color-font pairs is 4 × 6 = 24. Thus option C is correct. The answer 10 comes from adding 4 and 6, which does not enumerate complete designs. The value 18 is another unsupported sum or combination, while 36 is greater than the product and has no stated counting interpretation. Because both choices are required and are independently available, multiplication counts every valid design exactly once.

Each of the 2 bats can be paired with any of the 7 balls. Therefore, the total number of choices is 2 × 7 = 14.

For each of the 9 letter styles, either of the 2 border colours can be chosen. Therefore, by the fundamental principle of counting, the total number of ways is \(9 \times 2=18\). Adding the numbers to get \(11\) is incorrect because one style and one colour are chosen together. Exam tip: Multiply the numbers of choices when independent selections are made together.

Each of the 5 chapters can be combined with any of the 6 difficulty levels. By the fundamental counting principle, there are 5 × 6 = 30 choices.

The fundamental principle of counting says that when a complete outcome is formed through successive independent choices, the numbers of choices are multiplied. Selecting an entry gate gives 3 possibilities. For each selected gate, the seating room can be chosen in 4 ways. Therefore the number of gate-room pairs is 3 × 4 = 12, so option C is correct. The result 7 comes from adding the two numbers, which would not count a pair consisting of both a gate and a room. The values 9 and 16 do not follow from the stated two-stage process. The multiplication includes every possible gate with every possible room and therefore gives the exact number of valid selections.

For each of the 4 colours, any one of the 3 wheel designs can be chosen. Therefore, by the fundamental principle of counting, the total number of models is \(4 \times 3 = 12\). Adding to get \(4+3=7\) is incorrect because colour and wheel design are independent choices, and every combination is possible. Exam tip: multiply the numbers of choices in independent successive steps.

For each of the 6 heading colors, any of the 5 button shapes can be chosen. By the multiplication principle, the total is 6 × 5 = 30 ways.

The direct answer is option B: 20 ways. There are 4 possible boys for the boy representative and, independently, 5 possible girls for the girl representative. The fundamental principle of counting says that when one choice is made together with another choice, multiply the numbers of possibilities. Thus, total ways are \(4\times5=20\). Option A, 9, comes from adding 4 and 5; addition would describe a choice of a boy or a girl, not one of each. Option B, 20, is correct because every one of the 4 boys can be paired with each of the 5 girls. Option C, 16, has no valid counting basis here and is not \(4\times5\). Option D, 25, incorrectly uses \(5\times5\), although only 4 boys are available. A useful exam cue is: if all required selections must be made, multiply the choices for the separate stages.

Each of the 3 cover designs can be paired with any of the 9 page colors. By the multiplication principle, there are 3 × 9 = 27 ways.

There are 8 choices for the sweet type, and for each selected type there are 2 choices of box size. By the fundamental principle of counting, the total number of choices is \(8\times2=16\). The options 10, 8, and 12 do not correctly multiply the two independent choices. Exam tip: Multiply the number of choices when successive selections are independent.

There are 6 choices of pen colour, and for each chosen pen colour there are 3 choices of highlighter colour. By the fundamental principle of counting, the total number of pairs is \(6 \times 3=18\). Option 9 results from adding the choices, but choices made from two separate groups are multiplied, not added. Exam tip: Multiply the number of choices at successive independent stages.

The governing concept is the fundamental principle of counting, also called the multiplication rule. The selection has two successive and independent stages: first choose one file color, and then choose one label color. The file color can be selected in 4 ways. For each of these 4 choices, any of the 7 label colors can be paired with it, so each file-color choice creates 7 complete pairs. Therefore, the total number of distinct choices is 4 × 7 = 28. Option B is correct. Adding 4 and 7 gives 11, but addition is appropriate for alternatives, not for completing both selections. Options C and D do not equal the required product.

Each of the 5 teachers can be paired with any of the 4 time slots. Thus, by the multiplication principle, there are 5 × 4 = 20 ways.

Each of the 2 card types can be paired with any of the 8 validity plans. The multiplication principle gives 2 × 8 = 16 ways.

The direct answer is option A: 18 identities. An identity is formed by selecting exactly one team name and exactly one jersey color. There are 3 choices for the name. For each chosen name, there are 6 possible colors, so the multiplication principle gives \(3\times6=18\). Option A is correct because every name can be combined with every color. Option B, 9, does not result from the required calculation; adding the categories would also be inappropriate. Option C, 12, is another incorrect product or partial count and does not include all name-color pairs. Option D, 36, reverses no useful rule but doubles the correct result; there are not 6 choices for both parts. The choices are independent, so a name does not reduce or change the number of available colors. Count a few examples if needed: name 1 has 6 identities, name 2 has 6, and name 3 has 6; hence \(6+6+6=18\), the same as \(3\times6\).

There are 7 choices for the playlist, and for each selected playlist there are 3 choices for sound quality. By the fundamental principle of counting, the total number of choices is \(7\times3=21\). Option 10 comes from adding the numbers, which is incorrect here because two independent selections are being made. Exam tip: Multiply when every choice of one step can be paired with every choice of the next step.

For each of the 5 brush sizes, any one of the 5 colour palettes can be selected. Therefore, by the fundamental principle of counting, the total number of selections is \(5 \times 5=25\). Adding the numbers gives 10, but addition is not used here because a brush size is paired with a palette. Exam tip: multiply the numbers of choices when independent selections are made together.