Fundamental principle of counting

Correct answer: C. 2520 codes

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.

Correct answer: A. 720 options

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.

Correct answer: B. 1344 numbers

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.

Correct answer: A. (126) selections

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.

Correct answer: C. outcomes

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.

Correct answer: B. (32) strings

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.

Correct answer: B. 132 options

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.

Correct answer: C. 600 signals

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.

Correct answer: D. 240 numbers

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.

Correct answer: A. (144) ways

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.

Correct answer: B. 36 codes

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.

Correct answer: C. 240

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.

Correct answer: D. 120 numbers

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.

Correct answer: B. 60

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.

Correct answer: A. 1080 passwords

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.

Correct answer: B. 210 selections

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.

Correct answer: C. 2700 ways

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.

Correct answer: B. 720 avatars

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

Correct answer: C. 1176 numbers

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.

Correct answer: B. 625 codes

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.

Correct answer: A. 54 cards

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.

Correct answer: A. 280 numbers

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.

Correct answer: B. 5400 PINs

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.

Correct answer: B. 504 outcomes

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.

Correct answer: C. 120 patterns

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.