Permutations

Correct answer: A. (144)

Arrange the boys around the circle in ((4-1)!) ways and place the girls in the (4) gaps in (4!) ways. For circular alternate seating, fix one group first.

Correct answer: B. (453600)

Here (E,T,I) are each repeated twice, so the count is (10!/(2!2!2!)). Divide by the factorial of each repeated letter.

Correct answer: D. (10800)

Inside the vowel block there are (5!/(2!2!)) arrangements, and outside there are (6!/2!) arrangements. Do not forget repeated letters when forming blocks.

Correct answer: C. (576)

The four odd digits occupy the four odd positions in (4!) ways, and even digits fill the rest in (4!) ways. Identify the type of positions first.

Correct answer: A. (181440)

For a bracelet, the count is ((10-1)!/2) because both rotation and reflection are identical. Remember the difference between bracelets and circular arrangements.

Correct answer: B. (1200)

The last digit is (0) or (5), and if (5) is last, (0) must be included inside. Count divisibility and compulsory digit cases separately.

Correct answer: D. (26880)

Among the (6) relative orders of (A,B,C), (A) is first in (2) orders, so the valid fraction is (4/6). Relative order solves such questions quickly.

Correct answer: C. (604800)

Arrange the men in (6!) ways and place the (4) women in (^{7}P_4) ways in the (7) gaps. Use the gap method for non-adjacent conditions.

Correct answer: A. (84)

In increasing order, each chosen set gives only one arrangement, and including (0) would not give a (6)-digit number. So choose (6) digits from (1) to (9).

Correct answer: D. (140)

Add arrangements starting with smaller available letters at each position, then add (1). In rank problems, first arrange letters alphabetically.

Correct answer: B. (44)

This is the derangement of (5) objects, whose value is (44). In derangement, no object remains in its correct place.

Correct answer: C. (15)

Choose the (2) wrong letters in (^{6}C_2) ways, and they must swap with each other. For exactly (2) wrong, those two letters interchange.

Correct answer: A. (3360)

Here (A) is repeated twice and (L) three times, so the count is (8!/(2!3!)). Count repeated letters first.

Correct answer: D. (360)

Treat the three (L)'s as one block, giving (6) units with (A) repeated twice, so the count is (6!/2!). A block of identical letters acts as one unit.

Correct answer: B. (1920)

There are (2) ends for (C), and for each, (8) ordered position pairs for (A,B), followed by (5!) arrangements. Fix the end and distance first.

Correct answer: C. (80640)

Treat the two specified letters as one block, choose (5) more letters from the remaining (8), and arrange (6) units. Include the (2!) internal orders of the block.

Correct answer: A. (1440)

The three odd positions can be filled from (4) odd digits in (^{4}P_3) ways, and the remaining three positions from (5) remaining digits in (^{5}P_3) ways. Apply position restrictions first.

Correct answer: D. (1663200)

The number of arrangements is (12!/(4!3!2!)). Divide by the factorial of each group of identical objects.

Correct answer: B. (30240)

Total circular arrangements are (8!), and adjacent cases are (2\cdot 7!), so the difference is (30240). In circular arrangements, start with ((n-1)!).

Correct answer: C. (554400)

It has (N) three times, (E) three times and (D) twice, so the count is (11!/(3!3!2!)). Write frequencies before applying the formula.

Correct answer: D. (35)

Place the (6) white balls first, creating (7) gaps, and choose (4) of them for black balls. With identical objects, only position selection is needed.

Correct answer: A. (14832)

Choose the correctly assigned person in (8) ways, and derange the remaining (7) people with (D_7=1854). Exactly one correct means all others are wrong.

Correct answer: B. (230400)

Choose the even and odd digit groups and count (7!) arrangements, then subtract leading-zero cases. When (0) is included, handle the first digit carefully.

Correct answer: C. (529200)

Arrange consonants in (7!/(2!2!)) ways, then choose (4) of the (8) gaps and arrange vowels in (4!/(2!2!)) ways. Use the gap method for non-adjacent vowels.

Correct answer: D. (252)

Any chosen set of (5) digits has only one decreasing arrangement, and the first digit will not be zero. So choose (5) digits from (10).