Permutations

Treat each couple as one block; the (4) blocks have ((4-1)!) circular arrangements and each block has (2!) internal orders. Exam tip: count the internal swap of every couple.

Choose the position of the special flag in (3) ways and fill the other (3) positions from (9) flags in (^{9}P_{3}) ways. Exam tip: count allowed positions of the compulsory object first.

Use inclusion-exclusion on total (8!), subtracting cases with (T) first and (E) last: (8!-7!-7!+6!). Exam tip: add back the overlap in two forbidden conditions.

Arrange the (5) mathematics books in (5!) ways, then place the (4) physics books in (^{6}P_{4}) gaps. Exam tip: gaps are the safest method for no two together.

First choose ordered seat pairs for (A,B) with distance at least (3); there are (42) such pairs, then seat the remaining (4) people in (^{6}P_{4}) ways. Exam tip: position-pair counting is useful for numbered seats.

Total arrangements are (\frac{7!}{2!}), and among the (4) relative orders of vowels only (1) order is allowed, so divide by (4!). Exam tip: divide by the factorial when relative order is fixed.

In total (7!) arrangements, the two relative orders of (A) and (B) are equally likely, so half are valid. Exam tip: use symmetry for relative order conditions like left of.

There are (2) alternate patterns, and in each pattern Indians can be arranged in (5!) ways and foreigners in (5!) ways. Exam tip: equal groups in alternate arrangement give (2) patterns.

Choose the other (4) digits with (0) and (8) in (\binom{7}{4}) ways; for each selection subtract (5!) leading-zero cases from (6!). Exam tip: leading zero correction is mandatory.

We assign (4) of the (6) rings to (4) distinct fingers with order, so (^{6}P_{4}=360). Exam tip: distinct fingers require permutations.

Fix (A); the opposite seat for (B) is fixed, and the remaining (8) people can be arranged in (8!) ways. Exam tip: fix one person to remove circular symmetry.

There are (11) letters; (I) occurs four times, (S) occurs four times, and (P) occurs twice, so the count is (\frac{11!}{4!,4!,2!}). Exam tip: recheck repeated-letter counts.

Choose the (2) correctly assigned prizes in (\binom{7}{2}) ways and derange the remaining (5), giving (!5=44). Exam tip: use derangement for exactly correct cases.

Choose the other (3) letters from (6), then subtract adjacent special-pair arrangements (2!\times4!) from total (5!). Exam tip: handle inclusion and adjacency restrictions separately.

The thousands digit must be from (5,6,7,8,9), and the units digit must be even; case-wise counting gives (400). Exam tip: count overlapping digit choices case by case.

Form an (AB) block among (8) objects, then subtract cases where (C) is adjacent to the block on either outside side; total is (2!\times(8!-2\times7!)). Exam tip: compound restrictions may need both block and subtraction.

There are (4) position pairs at distance (2) and (2!) orders for questions (1) and (2), and the remaining (4) questions arrange in (4!) ways. Exam tip: exactly one between means position difference (2).

Choose the end for (P) and (E) in (2) ways; the remaining (5) letters include (E) repeated twice, so there are (\frac{5!}{2!}) arrangements. Exam tip: divide for a repeated end-letter left inside.

Subtract adjacent-special-car cases (9\times2!\times{}^{8}P_{6}) from total (^{10}P_{8}). Exam tip: when parking spots are distinct, start from total arrangements.

First arrange the (5) white balls around a circle in ((5-1)!) ways, then place (3) black balls in (^{5}P_{3}) circular gaps. Exam tip: in circular gaps, arrange the base objects first.

Among total (7!) arrangements, the (3!) relative orders of (A,B,C) are equally likely, so (\frac{7!}{3!}) are valid. Exam tip: divide by factorial for fixed relative order.

Treat each couple as one block with fixed internal order, so only the (4!) blocks are arranged. Exam tip: do not multiply by (2!) when internal order is fixed.

Zero can occupy only the hundreds or tens place, giving (2) choices; the other (3) places are filled from (6) nonzero digits in (^{6}P_{3}) ways. Exam tip: decide the allowed positions of (0) first.

Treat (AB) and (CD) as two blocks; (6) objects can be arranged in (6!) ways and the blocks internally in (2!\times2!) ways. Exam tip: with multiple blocks, count each block's internal order.