Permutations

Correct answer: B. (384)

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.

Correct answer: B. (1512)

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.

Correct answer: B. (30960)

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.

Correct answer: A. (86400)

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.

Correct answer: B. (16800)

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.

Correct answer: D. (210)

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.

Correct answer: A. (2520)

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.

Correct answer: A. (28800)

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.

Correct answer: B. (13440)

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.

Correct answer: A. (360)

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

Correct answer: A. (80640)

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.

Correct answer: A. (34650)

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.

Correct answer: B. (1854)

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.

Correct answer: C. (2160)

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.

Correct answer: A. (400)

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.

Correct answer: D. (10080)

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.

Correct answer: B. (192)

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).

Correct answer: B. (240)

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.

Correct answer: B. (1451520)

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.

Correct answer: A. (2880)

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.

Correct answer: A. (840)

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.

Correct answer: A. (24)

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.

Correct answer: B. (300)

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.

Correct answer: A. (2880)

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.