Multiple Choice

1. A restaurant offers 3 appetizers, 5 main courses, and 2 desserts. How many different 3-course meals can be ordered?

   $3+5+2=10$

   $3! \times 5! \times 2!$

   $5 \times 2$

   $3 \times 5 \times 2 = 30$

2. A student can choose a novel from 12 English novels or 8 French novels. How many choices does the student have?

   $12 - 8 = 4$

   $12 + 8 = 20$

   $12 \times 8 = 96$

   $12! + 8!$

3. In how many different ways can 6 distinct books be arranged on a shelf?

   $6 \times 5 = 30$

   $6$

   $6! = 720$

   $6^2 = 36$

4. How many different 3-letter codes can be formed using the first 8 letters of the alphabet, if no letter is repeated?

   $P(8,3) = \frac{8!}{(8-3)!} = 8 \times 7 \times 6 = 336$

   $8^3 = 512$

   $8! = 40320$

   $8 \times 3 = 24$

5. How many 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, 7 if no digit is repeated?

   $P(7,4) = \frac{7!}{(7-4)!} = 7 \times 6 \times 5 \times 4 = 840$

   $7^4 = 2401$

   $7 \times 4 = 28$

   $7! = 5040$

6. A license plate consists of 3 letters followed by 3 digits. How many unique license plates are possible if repetition of letters and digits is allowed? (Assume 26 letters and 10 digits from 0-9).

   $(26+10)^3$

   $26 \times 3 + 10 \times 3$

   $(26 \times 10)^3$

   $26^3 \times 10^3 = 17,576,000$

7. In how many distinct ways can the letters of the word "MATH" be arranged?

   $4$

   $4! = 24$

   $4^4 = 256$

   $4 \times 3 = 12$

8. From a group of 10 students, a president, a vice-president, and a treasurer are to be chosen. In how many ways can this be done?

   $10 \times 3 = 30$

   $P(10,3) = \frac{10!}{(10-3)!} = 10 \times 9 \times 8 = 720$

   $C(10,3) = \frac{10!}{3!7!} = 120$

   $10^3 = 1000$

9. A student needs to choose one history book and one science book. There are 5 history books on the shelf, and 4 physics books and 3 chemistry books in the science section. How many different combinations of books can the student choose?

   $5+4+3=12$

   $5 \times 4 \times 3 = 60$

   $(5+4) \times 3 = 27$

   $5 \times (4+3) = 35$

10. How many 5-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, 7 if the number must be odd and no digit can be repeated?

    $5 \times 4 \times 3 \times 2 \times 1 = 120$

    $7! = 5040$

    $4 \times P(6,4) = 4 \times (6 \times 5 \times 4 \times 3) = 1440$

    $P(7,5) = 2520$

11. How many distinct arrangements of the letters of the word "ORANGE" are possible if all the vowels must always stay together?

    $720$

    $120$

    $144$

    $72$

12. How many $4$-digit numbers can be formed using the digits $1, 2, 3, 4, 5, 6, 7, 8$ if no digit is repeated and the number must be greater than $5000$?

    $P(8,4)$

    $1680$

    $840$

    $4 \times 7!$

13. In how many ways can $5$ distinct boys and $4$ distinct girls be seated in a row such that no two girls sit together?

    $15120$

    $9!$

    $5! \times 4!$

    $43200$

14. How many distinct $8$-letter arrangements can be made from the letters of the word "EQUATION" if the arrangement must start with a vowel and end with a consonant?

    $10800$

    $15120$

    $40320$

    $7200$

15. A set of $5$ distinct mathematics textbooks and $3$ distinct physics textbooks are to be arranged on a shelf. How many distinct arrangements are possible if all the mathematics textbooks must be kept together?

    $120$

    $8!$

    $720$

    $2880$

16. How many permutations of the letters of the word "GARDEN" are there such that neither the first letter nor the last letter is a vowel?

    $720$

    $288$

    $672$

    $P(6,2) \times 4!$

17. Seven students, including Sarah and John, are to be seated in a row for a photograph. In how many distinct ways can they be arranged if Sarah must sit to the immediate left of John?

    $6!$

    $2520$

    $7!$

    $P(7,2)$

18. How many distinct $4$-digit numbers can be formed using the digits $1, 2, 3, 4, 5, 6, 7$ without repetition, such that the number contains at least one even digit?

    $840$

    $816$

    $P(7,4) - P(7,0)$

    $P(4,4)$

19. Three distinct Mathematics books and three distinct English books are to be arranged on a shelf. How many distinct arrangements are possible if no two Mathematics books are adjacent?

    $3! \times 3!$

    $6! - (4! \times 3!)$

    $144$

    $6!$

20. A security code consists of $3$ distinct letters followed by $3$ distinct digits. The letters must be chosen from the first $8$ letters of the alphabet (A-H), and the digits must be chosen from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Additionally, the first letter must be a vowel, and the last digit must be odd. How many such distinct security codes are possible?

    $2 \times 7 \times 6 \times 9 \times 8 \times 5$

    $2 \times P(7,2) \times P(9,3)$

    $P(8,3) \times P(9,3)$

    $23520$