Fundamental principle of counting

The first three places can be filled with distinct vowels in 5 × 4 × 3 ways. The last two places can be filled with distinct consonants in 7 × 6 ways. Thus, the total is 5 × 4 × 3 × 7 × 6 = 2520 codes.

One choice is made from each of the five independent categories. By the multiplication principle, the total is 6 × 5 × 4 × 3 × 2 = 720 test options.

The thousands digit must be 6, 7, 8, or 9, so it has 4 choices. After choosing it, the other three places have 8, 7, and 6 choices, respectively, because digits cannot repeat. The total is 4 × 8 × 7 × 6 = 1344.

The direct answer is option A, 126 selections. The fundamental principle of counting says that if one choice can be made in 6 ways, a second in 7 ways, and a third in 3 ways, and the choices are made together, multiply the numbers. Thus total selections=6×7×3=42×3=126. Each selection is identified by one chapter, one question from the seven questions, and one difficulty level, so no addition is needed. Option A, 126, is correct. Option B, 96, does not result from the required product. Option C, 84, is 6×7×2 and ignores one level, while option D, 63, is 7×3×3 and does not use the six chapter choices correctly. The choices are not being arranged or repeated in a way that requires permutations; this is a simple multiplication principle question. Memory cue: independent ‘and’ choices are multiplied; alternative ‘or’ choices are usually added.

The governing concept is the fundamental principle of counting for successive independent experiments. Each coin has two possible results, so three distinguishable coin tosses produce 2 × 2 × 2 = 2^3 = 8 outcomes. Each die has six possible faces, so two distinguishable dice produce 6 × 6 = 6^2 = 36 ordered outcomes. Since the coin results and die results occur together, multiply the two counts: 2^3 × 6^2 = 8 × 36 = 288. Thus option C is correct. The order of the dice outcomes matters in the usual outcome space, although the product count gives the same total. Option A, B, or D results from an incorrect multiplication or from using the wrong number of outcomes for coins or dice.

This uses the fundamental principle of counting with fixed and free positions. A 7-place binary string has positions 1 through 7. The first position is fixed as 1, and the seventh position is also fixed as 1, so neither contributes a choice. The remaining five positions, 2 through 6, can independently contain either 0 or 1. Thus the number of strings is 2 × 2 × 2 × 2 × 2 = 2⁵ = 32. Therefore, option B is correct. Option C incorrectly treats all six nonfixed positions as free, option D ignores both fixed conditions, and option A counts only four free positions.

There are 6 × 8 = 48 possible packet-label pairs before restrictions. Excluding the 4 disallowed pairs leaves 44 valid pairs. Each has 3 size choices, so 44 × 3 = 132 valid options.

The first place has 5 choices. The second and third places must use different shapes, giving 6 × 5 choices. The fourth place has 4 choices. Thus, the total is 5 × 6 × 5 × 4 = 600 signals.

The first two places can be filled in 4 × 3 ways, and the last two in 5 × 4 ways. The two digit sets do not overlap, so the total is 4 × 3 × 5 × 4 = 240 numbers.

The governing concept is the fundamental principle of counting. The three selections are made from separate categories: the boy monitor can be chosen in 8 ways, the girl co-monitor in 6 ways, and the house in 3 ways. These choices are independent, and the order of making the selections does not change the resulting assignment. Hence the total number of ways is 8 × 6 × 3 = 48 × 3 = 144. Therefore, option A is correct. The other choices arise from omitting one factor, adding instead of multiplying, or using an incorrect product. No permutation formula is needed because each role already specifies the category from which its choice is made.

With M fixed first, the last position can use any of the 6 letters other than M and N. After choosing the last letter, 6 letters remain for the middle position. Thus, there are 6 × 6 = 36 codes.

For each model, one of 4 colours, one of 6 storage options, and one of 2 warranty plans can be selected. By the fundamental principle of counting, the total number of options is \(5 \times 4 \times 6 \times 2=240\). Hence, option C is correct. A result such as 360 would come from incorrect multiplication or double-counting a choice. Exam tip: When choices are made in independent stages, multiply the number of choices at each stage.

There are 3 choices for the hundreds digit, 10 for the tens digit, and 4 for the units digit. By the fundamental counting principle, the total is 3 × 10 × 4 = 120 numbers.

The choices of language, project, and presentation mode are independent. By the fundamental principle of counting, the total number of ways is \(4 \times 5 \times 3=60\). Therefore, 60 is the correct option. Adding \(4+5+3\) would be incorrect because the student makes one choice from each of the three categories. Exam tip: Multiply the numbers of choices when selections are made independently in successive stages.

The first place has 6 choices and the second has 5 because it must differ from the first. The third place has 6 choices, and the equal fourth-and-fifth pair has 6 choices. Thus, 6 × 5 × 6 × 6 = 1080 passwords.

There are 7 choices for the main subject, then 6 remaining choices for the supporting subject, and 5 for the extra subject. Since the roles matter, the total is 7 × 6 × 5 = 210 selections.

For the outward journey there are 3 × 5 × 4 choices. On the return, there are 3 choices for the D–C leg because the outward C–D train cannot be reused, 5 for C–B, and 3 for B–A. The total is 3 × 5 × 4 × 3 × 5 × 3 = 2700 round trips.

Each feature can be chosen independently, so the number of avatars is 5 × 4 × 3 × 2 × 6 = 720.

The units digit must be one of 1, 3, 5, or 7, giving 4 choices. After that, the thousands digit has 7 nonzero choices left. The hundreds and tens digits then have 7 and 6 choices, respectively. Thus, 4 × 7 × 7 × 6 = 1176 numbers.

Choose the first, second, third, and fourth letters freely, with 5 choices for each. The sixth letter is then fixed by the first, and the fifth by the second. Therefore, there are 5⁴ = 625 codes.

Without the restriction, there are 4 × 5 × 3 = 60 combinations. The restriction rules out 1 card type paired with 2 colors and any of the 3 validity periods, or 1 × 2 × 3 = 6 combinations. Thus, 60 − 6 = 54 valid cards are possible.

The hundreds digit must be 3, 4, 5, 6, or 7, giving 5 choices. The tens digit then has 8 choices and the units digit has 7, since digits cannot repeat. Thus, 5 × 8 × 7 = 280 numbers.

The first place has 6 choices. The second has 5 choices because it must differ from the first. Each of the two middle places has 6 choices, and the last has 5 choices because it must differ from the first. Thus, the total is 6 × 5 × 6 × 6 × 5 = 5400 PINs.

The prizes are distinct, so order matters. There are 9 choices for first prize, 8 remaining choices for second, and 7 for third. Hence, the number of outcomes is 9 × 8 × 7 = 504.

Each question can be answered in the stated number of ways, and a choice is made for every question. By the multiplication principle, the number of answer patterns is 3 × 4 × 5 × 2 = 120.