Permutations

Correct answer: B. (4320)

Subtract arrangements where the (3) special books form one block, that is (5!\times3!), from total (7!). Exam tip: for not-together cases, subtract unfavorable cases.

Correct answer: B. (1260)

There are (7) letters and (R) repeats twice, so the count is (\frac{7!}{2!}). Exam tip: divide by factorials of repeated letters.

Correct answer: A. (10080)

Treat (TT) as one block, giving (8) objects with (M) and (E) repeated, so the count is (\frac{8!}{2!,2!}). Exam tip: form the block first, then handle repetitions.

Correct answer: C. (300)

The last digit must be (0) or (5); counting both cases separately gives (300). Exam tip: never allow zero in the first place.

Correct answer: B. (70)

Each selection of (4) digits has exactly (1) increasing arrangement, so the count is (\binom{8}{4}=70). Exam tip: when order is fixed, selection alone is counted.

Correct answer: C. (60480)

There are (5) position pairs for (P) and (Q), with (2!) orders, and the remaining (7) people can be arranged in (7!) ways. Exam tip: count distance conditions by position pairs.

Correct answer: A. (604800)

Arrange (6) men in (6!) ways, then place (4) women in (^{7}P_{4}) of the (7) gaps. Exam tip: use the gaps method for non-adjacent placement.

Correct answer: B. (907200)

Treat (MM) as one block; now there are (10) objects with (A) and (T) repeated twice, so the count is (\frac{10!}{2!,2!}). Exam tip: after blocking, do not forget remaining repetitions.

Correct answer: B. (2688)

Subtract unfavorable cases where both special objects are selected, (^{8}P_{2}\times{}^{4}P_{2}), from total (^{10}P_{4}). Exam tip: at most one means not both.

Correct answer: A. (44)

This is the derangement of (5) objects, so the number is (!5=44). Exam tip: identify derangement when no object goes to its own position.

Correct answer: C. (72)

Total circular arrangements are ((6-1)!), and adjacent (A,B) arrangements are (2!\times4!). Exam tip: for not adjacent, subtract the adjacent case from total.

Correct answer: D. (120)

Treat vowels (U,E) as one block; among (5) objects, (S) and (C) repeat twice, giving (\frac{5!}{2!,2!}\times2!). Exam tip: count the internal order of the vowel block separately.

Correct answer: B. (720)

The last place has (4) choices from (1,3,5,7), the first place then has (6) nonzero choices, followed by (6) and (5) choices. Exam tip: handle last digit and first digit restrictions first.

Correct answer: A. (20160)

For a necklace, both rotation and reflection are identical, so the count is (\frac{(9-1)!}{2}). Exam tip: remember the difference between circular arrangements and necklaces.

Correct answer: C. (15)

Choose the (4) correct letters in (\binom{6}{4}) ways, and derange the remaining (2) letters in (!2=1) way. Exam tip: split exact conditions into fixed and deranged parts.

Correct answer: A. (38880)

Subtract the cases where all (5) vowels form one block, (4!\times5!), from total (8!). Exam tip: not all together means not in one single block.

Correct answer: B. (1693440)

Arrange the boys in (7!) ways and place the (3) girls in (^{8}P_{3}) of the (8) gaps. Exam tip: permute separated objects into gaps.

Correct answer: C. (55440)

Choose the other (4) people from (11) in (\binom{11}{4}) ways and arrange the (5) people in (5!) ways. Exam tip: include the compulsory object first.

Correct answer: A. (19958400)

There are (11) letters, with (B) and (I) repeated twice, so the count is (\frac{11!}{2!,2!}). Exam tip: divide by the factorial of each repeated letter.

Correct answer: C. (30240)

Place the two special players in the (6) inner positions in (^{6}P_{2}) ways and arrange the remaining (6) players in (6!) ways. Exam tip: handle restricted positions first.

Correct answer: A. (17280)

Treat the (4) blue balls as one block, giving (6!) outside arrangements and (4!) internal arrangements. Exam tip: count internal orders for distinct objects inside a block.

Correct answer: C. (4560)

Group digits by residues modulo (3), count (5)-digit selections whose sum is divisible by (3), then arrange each selection in (5!) ways. Exam tip: use digit sum for divisibility by (3).

Correct answer: A. (720)

Treat (ABC) as one block with fixed internal order, so the circular arrangement of (6) objects is ((6-1)!). Exam tip: if internal order is fixed, do not multiply by an internal factorial.

Correct answer: B. (40320)

There are (8) adjacent chair pairs and (2!) orders for the two special students, then the remaining (4) students occupy (7) chairs in (^{7}P_{4}) ways. Exam tip: with distinct chairs, account for empty chairs too.

Correct answer: C. (6652800)

There are (12) letters; (N) and (E) occur four times each and (D) occurs twice, so the count is (\frac{12!}{4!,4!,2!}). Exam tip: correct repetition counts are crucial.