Fundamental principle of counting

Correct answer: C. (1440)

Fix one particular person; then the other particular person has (2) valid positions. Arrange the remaining (6) persons in (6!) ways.

Correct answer: C. (720)

Place (3) of the (4) odd digits in positions (1,3,5) in order. Then fill the even positions from the remaining digits.

Correct answer: C. (2460)

First choose (2) doctors, then choose the remaining (5) members from engineers and teachers with at least (2) engineers. Keep exact and at least conditions in separate stages.

Correct answer: C. (2160)

Treat all vowels as one block and account for repeated consonants. Count the arrangements inside the vowel block separately.

Correct answer: C. (4440)

Subtract arrangements containing the block (12) from total (^{7}P_6). Only the fixed order (1) immediately before (2) is forbidden.

Correct answer: C. (74)

First count positive solutions and subtract solutions where the eldest child gets (5) or more coins. Complement counting is useful for upper-bound restrictions.

Correct answer: C. (4116)

The first position has (3) choices, the last has (4), and each middle position has (7) choices. Multiply choices for independent positions.

Correct answer: A. (604800)

Among total (10!) arrangements, the (3!) relative orders of the three particular delegates are equally likely. Hence only one order is counted.

Correct answer: C. (240)

First choose (3) rows and (3) columns, then match them in (3!) ways. Matching is useful for row-column restrictions.

Correct answer: B. (505)

A function from a 4-element set to a 5-element set assigns one target element to each of the four source elements. For each source element there are 5 choices, and the choices are independent, so the total number of functions is \(5^4\). A one-one function uses different target elements for different source elements; its count is the permutation number \({}^{5}P_4\). Subtracting gives the functions that are not one-one.

Calculate the total: \(5^4=625\). Calculate the injective functions: \({}^{5}P_4=5\times4\times3\times2=120\). Hence the number that are not one-one is \(625-120=505\). Therefore option B is correct. The result is not found by subtracting only one or two arrangements; every non-injective function is included in the difference between all functions and one-one functions.

Correct answer: B. (123480)

The letters have (7^3) choices. For the sum of three distinct digits to be odd, the number of odd digits must be (1) or (3).

Correct answer: C. (26640)

First choose exactly (2) poetry books, then choose the remaining (3) books from novels and dramas with at least (1) novel. After selection, arrange in (5!) ways.

Correct answer: C. (43200)

First seat the men around the circle in ((6-1)!) ways and place the (4) women in the (6) gaps. Use the gap method for circular separation.

Correct answer: C. (1288)

Take cases for the even units digit because the thousands digit must be from (5) to (9). Decide the leading and last positions together.

Correct answer: C. (450)

For the sum to be odd and the product to be even, exactly (1) odd and (2) even numbers must be chosen. Classify parity conditions first.

Correct answer: C. (108000)

There are (5) possible positions for the block of three digits, then arrange digits and letters in order. Treat consecutive objects as a block.

Correct answer: B. (1440)

The only possible size distributions are ((2,2,1,1)) and ((2,2,2,0)). Count team choices and project assignments for each pattern.

Correct answer: C. (28800)

There are two possible patterns, starting with a boy or starting with a girl. In each pattern, boys and girls are arranged separately in (5!) ways.

Correct answer: B. (5040)

First divide (8!) by (3!) for the relative order of (A,B,C), then subtract cases where (D) is at an end. Relative orders are equally distributed.

Correct answer: B. (2736)

First consider (6)-digit sets that can be split into two (3)-digit parts with equal sums, then count arrangements inside both parts. For equal-sum conditions, think of sets and partitions first.

Correct answer: A. (20)

First choose the (2) children who sit correctly, then derange the remaining (3) children. Exact fixed-position conditions use derangements.

Correct answer: C. (19)

Place the (4) zeros in the (4) gaps around the (3) ones, with no gap receiving more than (2) zeros. Control forbidden consecutiveness by gap limits.

Correct answer: C. (33600)

First form the committee in (\binom{8}{3}\binom{6}{3}) ways, then choose (2) distinct posts from (6) members in order. Committee selection and post selection are separate stages.

Correct answer: A. (1440)

The third book has (2) end choices, and the two adjacent books are treated as a block with the remaining objects. Count the (2!) internal orders of the block too.

Correct answer: A. (420)

Choose the other (3) team members from (8), then choose the ordered speakers from those (3) members. Fix the compulsory but ineligible member first.