Permutations

Correct answer: A. (720)

There are (3) position pairs at distance (5) for (S,Q), with (2) orders, and the remaining (6) places fill in (6!/3!) ways. For gap-distance questions, count position pairs first.

Correct answer: C. (362880)

Treat the four specified people as one block with fixed internal order, so (9) units arrange in (9!) ways. If internal order is prescribed, do not multiply by (4!).

Correct answer: D. (168000)

First count arrangements containing both (0) and (9), then subtract cases where they are adjacent. When (0) is included, handle the leading position carefully.

Correct answer: B. (10800)

There are (5) odd positions among (9); place the (3) vowels in (^{5}C_3\cdot 3!/2!) ways and consonants in (6!/2!) ways. Decide position types first.

Correct answer: A. (1440)

Fix (A); then (B) and (C) occupy the two equally spaced positions in (2!) ways. The remaining (6) people sit in (6!) ways.

Correct answer: C. (3216)

By inclusion-exclusion, the count is (7!-3\cdot 6!+3\cdot 5!-4!). For multiple forbidden positions, this method is safe.

Correct answer: D. (7560)

Treat the two (C)'s as a block, giving (8!/(2!2!)) arrangements, then subtract the cases with adjacent (A)'s, (7!/2!). For two conditions, first form the required block.

Correct answer: B. (18635)

Add arrangements beginning with smaller available letters at each position and then add (1). Write the alphabetical order first when finding rank.

Correct answer: A. (2940)

The order is fixed within the chosen positions, but the first digit cannot be (0). In such restrictions, check both order and leading zero.

Correct answer: C. (14400)

For (A,B), use the (6) middle positions in (^{6}P_2) ways, then subtract adjacent cases (5\cdot 2\cdot 6!). Count middle-position and non-adjacent restrictions together.

Correct answer: D. (56)

Only the (3) vowel positions must be chosen, because both vowel and consonant orders are fixed. When orders are fixed, position selection gives the answer.

Correct answer: B. (3386880)

Total circular arrangements are (10!), and all-three-together cases are (8!\cdot 3!). For not-all-together cases, subtract the bad cases from total.

Correct answer: A. (3600)

Inside the vowel block there are (5!/(2!3!)) arrangements, and outside there are (6!/2!) arrangements. Handle the block and repetitions together.

Correct answer: C. (3840)

Treat each couple as a block; the (5) blocks arrange in (5!) ways and each block has (2) internal orders. Couples contribute (2^5) internal arrangements.

Correct answer: D. (8400)

Choose one of the two specified letters in (2) ways, choose (4) more from the remaining (7), and arrange in (5!) ways. For exactly-one conditions, count both choices separately.

Correct answer: B. (6300)

First include exactly one digit from (2,4,6), then make cases for an odd unit digit. Handle oddness and compulsory choice together.

Correct answer: A. (9979200)

Here (T) appears three times and (O,N,I) appear twice each, so the count is (12!/(3!2!2!2!)). For long words, make a frequency table.

Correct answer: C. (604800)

Among the relative orders of (A,B,C), (A) is in the middle with probability (1/3), and (D) before (E) has probability (1/2). Hence the count is (10!/6).

Correct answer: D. (33600)

The first digit must be even but not (0), the last digit must be odd, and the middle (4) places are filled from remaining digits. Always check the leading zero restriction.

Correct answer: B. (228690)

First arrange red and blue balls in (^{11}C_5) ways, then choose (4) of the (12) gaps for green balls. With identical objects, position selection is the key.

Correct answer: A. (480)

Treat (A,B,C) as a block, so (6) units have (5!) circular arrangements, and internally (4) orders do not have (B) in the middle. Count internal orders separately.

Correct answer: C. (7420)

Choose the (2) correctly assigned people in (^{8}C_2) ways and derange the remaining (6) with (D_6=265). Exactly (2) correct means all others are wrong.

Correct answer: D. (1108800)

Here (S) appears three times and (I) appears three times, so the count is (11!/(3!3!)). Divide by the factorial of each repeated group.

Correct answer: B. (15120)

First take total arrangements with (6) before (8), (8!/2), then subtract half of the block arrangements where (2,4) are adjacent. Separate order and non-adjacency conditions.

Correct answer: A. (2880)

There are (4) even positions among (9), and the (4) consonants occupy them in (4!) ways, while vowels fill the rest in (5!) ways. For position restrictions, first count the positions.