Permutations

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.

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

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.