Fundamental principle of counting

Correct answer: B. 540 numbers

The last digit can be 2, 4, or 6, giving 3 choices. Once it is chosen, the first digit has 6 choices: any of the remaining non-zero digits. The second and third places then have 6 and 5 choices. Thus, the total is 3 × 6 × 6 × 5 = 540 numbers.

Correct answer: C. 240 designs

A design consists of one choice from each of the five categories. By the multiplication principle, the total is 5 × 4 × 3 × 2 × 2 = 240 designs.

Correct answer: B. 4200 codes

The three digit positions can be filled in 7 × 6 × 5 ways, since digits cannot repeat. The two letter positions can be filled in 5 × 4 ways. The total is 7 × 6 × 5 × 5 × 4 = 4200 codes.

Correct answer: C. 147 options

There are 7 ways to choose only a drink, 7 × 5 = 35 ways to choose a drink with a snack, and 7 × 5 × 3 = 105 ways to choose a complete set. These are separate types of choice, so the total is 7 + 35 + 105 = 147 options.

Correct answer: C. 180 plans

There are 5 × 4 × 3 = 60 ways to choose the bus, meal, and museum option. For the guide, there are 3 choices: either one of the 2 guides or no guide. Thus, there are 60 × 3 = 180 plans.

Correct answer: D. 432 codes

The first two places allow 3² choices, and the last two allow 2² choices. For the middle pair, there are 4 choices for the first place and 3 different remaining choices for the second. Thus, the total is 3² × 4 × 3 × 2² = 432 codes.

Correct answer: C. 630 numbers

A number is divisible by 5 if its last digit is 0 or 5. If the last digit is 0, the first digit has 8 choices, followed by 7 and 6 choices for the remaining places: 8 × 7 × 6 = 336. If the last digit is 5, the first digit has 7 choices (it cannot be 0), followed by 7 and 6 choices: 7 × 7 × 6 = 294. Thus, the total is 336 + 294 = 630.

Correct answer: A. 114

Without the restriction, the multiplication principle gives 5×6×4 = 120 packages, because one drink, one snack, and one dessert are selected independently. The forbidden condition fixes both the drink as coffee and the dessert as chocolate. The snack can still be any of the 6 snacks, so exactly 6 packages are invalid. Subtracting them gives 120 − 6 = 114 valid packages. Therefore option A is correct. The value 120 ignores the incompatibility, while 116 and 118 subtract the wrong number of forbidden combinations. The restriction removes only the coffee–chocolate pair, not every package containing either item.

Correct answer: A. 240

The governing concept is the fundamental principle of counting, with separate cases for the blocks because block C has a restriction. For block A, there are 12 possible rows and 8 possible columns, giving 12×8=96 codes. Block B gives another 12×8=96 codes. In block C, only the six even rows 2, 4, 6, 8, 10, and 12 are allowed, and each can be paired with any of the 8 columns, giving 6×8=48 codes. Since the three blocks are distinct, the cases are added: 96+96+48=240. Hence option A is correct. Multiplying 12×8×3 would give 288, but that incorrectly allows odd rows in block C. The restriction must be applied before adding the case totals.

Correct answer: C. 324

First count the background-font combinations. Without restrictions, there are 5 × 4 = 20 such pairs. Since 2 specified pairs are invalid, the number of valid pairs is 20 − 2 = 18. Each valid pair can be combined independently with any of 6 icons and any of 3 layouts, giving 6 × 3 = 18 choices for those two components. By the multiplication principle, the total number of valid themes is 18 × 18 = 324. Therefore option C is correct. Option A ignores the invalid pairs and gives 20 × 18. The other options subtract an incorrect quantity or mishandle the independent icon and layout choices.

Correct answer: A. 2 × 4 × 3 × 3 = 72

Use the fundamental principle of counting: when successive choices are independent, multiply the number of choices at each stage. The first section contributes 2 choices because one question is selected from 2. In the second section, order matters, so the number of ordered selections of 2 questions from 4 is 4P2 = 4 × 3 = 12. The third section contributes 3 choices. Hence the total is 2 × 12 × 3 = 2 × 4 × 3 × 3 = 72, so option A is correct. Option B incorrectly uses 4C2, option C omits the ordering factor, and option D counts four choices twice.

Correct answer: B. 1 × 2 × 5 + 3 × 3 × 5 = 55

The governing concept is the fundamental principle of counting with separate cases. The circular shape forms one case: it has 1 shape choice, 2 permitted colour choices, and 5 letter choices, giving 1 × 2 × 5 = 10 badges. The remaining three shapes form a second case: they have 3 shape choices, 3 colour choices, and 5 letter choices, giving 3 × 3 × 5 = 45 badges. These cases cannot overlap, so their totals are added: 10 + 45 = 55. Therefore, option B is correct. Option A wrongly gives the circular shape all three colours, while C gives every shape only two colours and D omits the circular case.

Correct answer: A. 3 × 2 × 5² = 150

The governing concept is the multiplication principle with a repetition restriction. For the first position, there are 3 choices. Once that choice is made, the second position cannot use it, so only 2 choices remain. The third position is independent and has 5 choices, and the fourth position is also independent with 5 choices. Thus the total is 3 × 2 × 5 × 5 = 150. Therefore, option A is correct. Option B incorrectly allows the first choice to be repeated in the second position. Option C gives the second position all five choices, ignoring the restriction, and option D counts only one five-choice position instead of two.

Correct answer: A. 6^3

The fundamental principle of counting says that when one choice is made independently at each stage, the total number of outcomes is found by multiplying the numbers of choices. For each fixed city, there are 2 hotel choices and 3 transport choices, giving 2 × 3 = 6 combinations. Since the order A, B, C is already fixed, there is no additional factor of 3! for arranging the cities. The six choices occur independently for each of the three cities, so the total is 6 × 6 × 6 = 6^3 = 216. Hence option A is correct; 18 adds choices instead of combining stages.

Correct answer: A. 240

The governing concept is the fundamental principle of counting: when independent choices are made in succession, the total number of outcomes is the product of the choices at each stage. From the first, second, and third sections there are 5, 4, and 6 choices. The fourth section must contribute an easy question, so only 2 choices are available, not all 5 questions. Therefore the number of selections is 5 × 4 × 6 × 2 = 20 × 12 = 240. Hence option A is correct. The difficult questions are excluded by the condition, so using 3 or 5 for the fourth factor would overcount; the other numerical options do not follow the required product.

Correct answer: A. 1406

The governing concept is the inclusion–exclusion principle applied to the fundamental counting rule. Since each of the five positions can contain any of five letters, the total number of unrestricted signals is 5⁵ = 3125. Signals missing P can use only Q, R, S and T, so there are 4⁵ = 1024; similarly, 4⁵ signals miss Q. Signals missing both P and Q use only three letters and number 3⁵ = 243. Therefore, the required count is 5⁵ − 4⁵ − 4⁵ + 3⁵ = 3125 − 2048 + 243 = 1320. Hence the supplied options do not contain the mathematically correct answer; option A must be corrected to 1320. The other choices cannot be selected because they do not follow the required inclusion–exclusion calculation.