Fundamental principle of counting

Correct answer: B. (320) पासवर्ड

The governing concept is the fundamental principle of counting: when successive positions are filled independently, multiply the number of choices for each position. The first letter has 4 choices, and because repetition is allowed, each of the second and third letter positions also has 4 choices. The final digit has 5 choices. Thus the total is 4 × 4 × 4 × 5 = 4^3 × 5 = 64 × 5 = 320. Therefore option B is correct. Option A would not count all three letter positions, option C uses an incorrect product, and option D does not apply the required three-letter-one-digit structure. The stated order means no additional rearrangement is needed.

Correct answer: C. (30) नियुक्तियां

This uses the fundamental principle of counting with a restriction. First choose the president: there are 6 possible students. After that appointment, the selected student cannot be secretary, so only 5 students remain eligible for secretary. Because the posts are different, choosing student X as president and student Y as secretary is different from the reverse assignment; therefore we multiply rather than divide by 2. The total is 6 × 5 = 30, so option C is correct. Option A incorrectly allows the same student to occupy both posts, option B adds choices instead of multiplying, and option D corresponds to an unrelated combination count.

Correct answer: A. (80) तरीके

The fundamental principle of counting applies because the choices are made in successive stages. First, there are 5 choices for the chapter. Once a chapter is selected, that chapter supplies 8 choices for the question. Finally, the student chooses one of 2 answer languages. Each complete outcome is determined by one chapter, one question from that chapter, and one language, so the counts are multiplied: 5 × 8 × 2 = 40 × 2 = 80. Therefore option A is correct. Option B omits the language choice, option C adds the numbers rather than multiplying them, and option D doubles the correct result without a stated additional choice.

Correct answer: A. (16) स्ट्रिंग

The governing concept is the multiplication principle with fixed positions. A six-place binary string has positions 1 through 6. The first position is forced to be 1, and the sixth position is also forced to be 1, so neither contributes a choice. The four middle positions, 2 through 5, may independently contain either 0 or 1, giving 2 choices for each. Therefore the total number is 2 × 2 × 2 × 2 = 2^4 = 16. Option A is correct. Option B counts five free positions, option C incorrectly treats the restrictions as reducing the count too far, and option D counts all six positions as unrestricted.

Correct answer: B. (30) चयन

The governing idea is ordered selection without repetition, which can be counted by the fundamental principle of counting. There are 6 choices for the main subject. After choosing it, the supporting subject must be different, so only 5 choices remain. Because the roles are distinct and order matters, choosing Mathematics as main and Physics as supporting is different from choosing Physics as main and Mathematics as supporting. Thus the total is 6 × 5 = 30, equivalently 6P2. Option B is correct. Option A is the unordered combination 6C2, option C incorrectly allows the same subject twice, and option D does not follow either the ordered or unordered calculation.

Correct answer: B. 60 patterns

The governing idea is the fundamental principle of counting: when a task consists of successive independent choices, the total number of outcomes is the product of the numbers of choices at each stage. The first answer can be selected in 4 ways, the second in 5 ways, and the third in 3 ways. Therefore, the number of complete answer patterns is 4 × 5 × 3 = 20 × 3 = 60. Option B is correct because every choice for the first question can be paired with every choice for the second and third questions. Option A adds only some possibilities, while 45 and 75 do not result from the required multiplication.

Correct answer: C. 120 designs

This uses the fundamental principle of counting. A complete certificate design is formed by making one choice from each independent category: title style, seal, border, and signature color. Thus the first component can be chosen in 4 ways, the second in 3 ways, the third in 5 ways, and the fourth in 2 ways. The total number is 4 × 3 × 5 × 2 = 12 × 10 = 120. Hence option C is correct. The value 60 omits one factor or combines the choices incorrectly; 90 and 240 also do not equal the product of all four independent choice counts. No arrangement formula is needed because the task is selecting one feature from each separate category.

Correct answer: A. 5⁴ = 625

The governing concept is the multiplication principle for independent assignments. Consider the letters one at a time. The first distinct letter can be placed in any of 5 envelopes. The second also has 5 choices, because an envelope may already contain another letter and empty envelopes are allowed. The third and fourth letters likewise have 5 choices each. Therefore the total number of assignments is 5 × 5 × 5 × 5 = 5⁴ = 625. Option A is correct. Option B reverses the roles of letters and envelopes and counts the wrong objects. Option C incorrectly treats the choices as decreasing, which would apply only if repetition or shared envelopes were forbidden. Option D adds choices instead of multiplying them.

Correct answer: C. 27

This problem uses the fundamental principle of counting: when a task has successive independent choices, multiply the numbers of choices. The hundreds place can contain any of 1, 5, or 9, so it has 3 choices. Because repetition is allowed, the tens place again has 3 choices, and the units place also has 3 choices. Therefore, the total number is 3 × 3 × 3 = 3³ = 27. Option C is correct. If repetition were forbidden, the count would be 3 × 2 × 1 = 6, which explains option D, but that condition is not given. The values 9 and 18 do not include all three independent positions with repetition permitted. None of the digits is zero, so every formed sequence is genuinely a 3-digit number.

Correct answer: C. 2⁶ - 1

For each of the 6 distinct coins, there are exactly two independent choices: include it in the selection or leave it out. By the fundamental principle of counting, the total number of subsets is therefore 2 × 2 × 2 × 2 × 2 × 2 = 2⁶. This total includes the empty selection, in which no coin is chosen. Removing that one case gives 2⁶ − 1 = 63 non-empty selections. Hence option C is correct. The expression 6! − 1 counts neither subsets nor selections correctly, because factorial concerns arrangements. ⁶P₁ counts only one-coin ordered choices, while ⁶C₂ counts selections of exactly two coins, not all possible non-empty sizes.