Fundamental principle of counting

Correct answer: B. (26\times25\times26\times100)

There are (26\times25) ways for uppercase letters, (26) for the lowercase letter, and (10^2) for digits. When positions are fixed, multiply independent choices.

Correct answer: A. (3\cdot2^5)

The first day has (3) choices and each later day has (2) choices different from the previous day. With consecutive restrictions, choices reduce after the first position.

Correct answer: A. (2296)

If the last digit is (0), there are (9\times8\times7) ways; if it is (2,4,6,8), there are (4\times8\times8\times7) ways. The total is (504+1792=2296).

Correct answer: B. (31)

Total patterns are (2^5), and the one pattern with all false is removed. For at least conditions, the complement method is often faster.

Correct answer: A. (21\cdot20\cdot19\cdot10\cdot9)

The order is fixed, so the choices are (21,20,19,10,9) respectively. With distinctness, choices decrease after each selection.

Correct answer: B. (40)

There are (5) choices for president and (8) remaining choices for secretary. When posts are different, order matters.

Correct answer: C. (48)

Languages and themes give (2\times4), and ordered security questions give (3\times2) ways. The total is (2\times4\times3\times2=48).

Correct answer: A. (648000)

The first three digits can be chosen in (9\times9\times8) ways and the last three in (10^3) ways. The total is (9\times9\times8\times1000=648000).

Correct answer: A. (26\cdot25\cdot500)

For (3) digits, (500) ordered triples have even sum because half of all (1000) triples give even sum. Multiply by (26\times25) for letters.

Correct answer: B. (1120)

The fundamental principle of counting says that when independent choices are made in sequence, the total number of outcomes is found by multiplying the number of choices at each stage. One question must be selected from each of four chapters, so the chapter choices are multiplied rather than added.

The first, third, and fourth chapters provide 5, 7, and 8 choices. The second chapter has 6 questions, but 2 are difficult and cannot be chosen, leaving \\(6-2=4\\) allowed choices. Thus the total is \\(5\\times4\\times7\\times8=1120\\). The restriction affects only the second factor; it does not remove choices from the other chapters. Therefore option B is correct.

Correct answer: A. (26\cdot25\cdot1900)

For at least one (5) in the first two digits, there are (10^2-9^2=19) ways and (100) ways for the remaining two digits. With letters, the total is (26\times25\times19\times100).

Correct answer: A. (100)

If the first option is (A), paths are (1\times5\times2=10); for the other (3) options, paths are (3\times5\times6=90). The total is (100).

Correct answer: A. (240)

One letter must repeat, so the repeated letter can be chosen in (4) ways and the positions arranged in (5!/2!) ways. The total is (4\times60=240).

Correct answer: A. (1344)

The first digit has (4) choices: (6,7,8,9). The remaining places have (8\times7\times6) ways, so the total is (1344).

Correct answer: A. (210)

Stripes are considered ordered, so there are (7\times6) ways and (5) choices for the logo. The total is (7\times6\times5=210).

Correct answer: A. (56\cdot3^5)

Choose the (3) questions with (A) in (\binom{8}{3}) ways, then each of the remaining (5) questions has (3) non-(A) choices. The total is (\binom{8}{3}3^5).

Correct answer: A. (36)

There are (4) choices for the first and last letter. The second and third positions have (3) and (3) choices respectively, so the total is (36).

Correct answer: A. (900000)

There are (9) choices for the first digit, (10^4) for the middle (4) positions, and (10) for the equal last pair. The total is (9\times10^4\times10=900000).

Correct answer: A. (20)

There are (5) choices for the first place and (4) for the second. For fixed-type positions, apply the multiplication rule directly.

Correct answer: A. (5\cdot4^5)

The first question has (5) choices and each next question has (4) choices different from the previous one. For consecutive restrictions, count position by position.

Correct answer: A. (196)

The hundreds place has (4) choices: (4,5,6,7). The remaining two places have (7\times7) ways, giving (196).

Correct answer: A. (600)

Symbols can be chosen in (6\times5) ways and colors in (5\times4) ways. The choices are independent, so the total is (600).

Correct answer: B. (500)

The first place has (4) choices and the remaining three places have (5^3) choices. The total is (4\times125=500).

Correct answer: B. (36)

The special option at level two gives (2\times1\times3=6) paths and the other (3) choices give (2\times3\times5=30) paths. The total is (36).

Correct answer: B. (602)

Total words are (3^6), words missing at least one letter are handled by inclusion-exclusion. The correct count is (729-3\cdot2^6+3=540).