Fundamental principle of counting

Correct answer: A. (32805)

If the first digit is (7), there are (9^4) ways; if (7) is in one of the other (4) places, there are (4\times8\times9^3) ways. The total is (9^4+4\cdot8\cdot9^3=29889).

Correct answer: B. (150)

Choose the positions of (6) in (\binom{4}{2}) ways, and the remaining (2) positions have (5^2) choices. The total is (\binom{4}{2}5^2=150).

Correct answer: A. (448)

The first strip has (8) choices and each of the next two strips has (7) choices. The third may match the first but not the second, so the correct count is (8\times7\times7=392).

Correct answer: B. (72)

There are (3) choices for the core subject, (4\times3) ordered choices for electives, and (2) choices for the club. The total is (3\times4\times3\times2=72).

Correct answer: A. (45360)

The first equal pair has (9) choices, and the last (4) distinct digits have (10\times9\times8\times7) ways. The total is (9\times10\times9\times8\times7=45360).

Correct answer: B. 480

The number of ordered selections of two questions from the first section is \(^{5}P_{2}=5\times4=20\). The second and third sections provide \(6\) and \(4\) choices respectively. By the multiplication principle, the total number of ways is \(20\times6\times4=480\). Option 240 would result from not treating the order of the first two questions as distinct. Exam tip: Use permutations, not combinations, whenever order matters.

Correct answer: A. (36)

The fundamental principle of counting says that when choices are made in separate compatible cases, we count each case and then add the results. For one font, each permitted color gives one font-colour combination. The restricted fonts and unrestricted fonts must therefore be counted separately because they do not have the same number of available colors.

There are 2 special fonts, each available in 3 colors, giving 2×3=6 combinations. The remaining 7-2=5 fonts are available in all 6 colors, giving 5×6=30 combinations. These cases are disjoint, so the total is 6+30=36. Option A is correct. Multiplying all 7 fonts by all 6 colors would incorrectly include unavailable combinations for the special fonts.

Correct answer: A. (180)

The last digit can be chosen from (3,5,7) in (3) ways. The remaining places have (5\times4\times3) ways, giving (180).

Correct answer: B. (5)

The direct answer is option B, 5. A true-false answer pattern has two choices at every question: True or False. However, the first answer is already fixed as False, so it does not need to be chosen. Exactly four answers must be True. They must therefore be selected from questions 2, 3, 4, 5 and 6, which are five questions. Choose the one question that will be False, or equivalently choose four True answers: \binom{5}{4}=5. Thus there are 5 patterns. Option A, 10, would come from an incorrect selection count and does not respect the exact condition. Option B, 5, is correct because it counts the possible placements of four True answers among the remaining five positions. Option C, 15, overcounts the patterns, perhaps by using an unsuitable combination. Option D, 20, is also too large and ignores that the first position is fixed. Memory cue: fix the forced answer first, then count only the unfixed positions.

Correct answer: A. (16)

The direct answer is option A, 16. The first step is fixed as option 1, so it contributes only one possibility. At every later step, the option used immediately before cannot be repeated. Since there are three options in total, two choices remain at each of steps 2, 3, 4 and 5. The multiplication principle gives \(1\times2\times2\times2\times2=2^4=16\). Option A, 16, is correct. Option B, 24, would incorrectly treat the later choices as if three choices were always available. Option C, 32, also overcounts by giving too many choices after the first step. Option D, 48, is another overcount and does not apply the consecutive-repetition restriction. Notice that a choice may repeat after one intervening step; only adjacent equal choices are forbidden. Memory cue: fixed first step gives 1, and each following step gives 2.

Correct answer: A. (6720)

Symbols can be arranged in (5\times4) ways and letters in (8\times7\times6) ways. The total is (5\times4\times8\times7\times6=6720).

Correct answer: A. (51)

Sleeper gives (1\times3\times2=6) ways and the other coaches give (3\times3\times5=45) ways. The total is (51).

Correct answer: C. (80)

The thousands place has (2) choices: (1) or (2). The remaining (3) places have (5\times4\times3) ways, so the total is (120).

Correct answer: A. (21\cdot26^2\cdot4)

The first letter has (21) consonant choices, the next two letters have (26^2) choices, and the special symbol has (4) choices. The total is (21\cdot26^2\cdot4).

Correct answer: A. (100)

Premium effects give (5\times2\times2=20) combinations and the remaining effects give (5\times4\times4=80). The total is (100).

Correct answer: C. (240)

One digit appears twice and the others appear once each. The repeated digit has (4) choices and arrangements are (5!/2!), giving (240).

Correct answer: A. (60)

After excluding player (A), (5) players remain and medals are ordered distinct positions. The outcomes are (5\times4\times3=60).

Correct answer: A. (8\cdot7\cdot6\cdot3\cdot9^2)

The letters can be chosen in (8\times7\times6) ways and the position of (0) can be chosen in (3) ways. The remaining two digits have (9^2) choices.