Multiple Choice

1. What is the value of $\binom{10}{3}$?

   $30$

   $120$

   $720$

   $100$

2. A committee of $4$ people is to be chosen from a group of $9$ people. How many different committees can be formed?

   $3024$

   $36$

   $84$

   $126$

3. From a list of $12$ books, how many ways can you choose $5$ books to read?

   $95040$

   $60$

   $792$

   $495$

4. How many ways can you choose $2$ cards from a standard deck of $52$ cards?

   $2652$

   $1326$

   $51$

   $78$

5. A pizza shop offers $15$ different toppings. If you want to choose $4$ toppings for your pizza, how many different combinations of toppings are possible?

   $910$

   $1365$

   $60$

   $32760$

6. A committee of $5$ people is to be formed from a group of $8$ men and $6$ women. If a specific man and a specific woman must be on the committee, how many different committees can be formed?

   $100$

   $360$

   $220$

   $792$

7. A school needs to select $3$ students for a competition from a class of $20$. If $2$ particular students are ineligible to be selected, how many different groups of $3$ students can be chosen?

   $816$

   $4896$

   $1140$

   $612$

8. A restaurant offers $6$ main courses, $5$ side dishes, and $3$ desserts. How many ways can a customer choose $1$ main course, $2$ side dishes, and $1$ dessert?

   $120$

   $240$

   $90$

   $180$

9. There are $7$ points on a plane, no three of which are collinear. How many distinct lines can be drawn by connecting any two points?

   $35$

   $42$

   $14$

   $21$

10. There are $8$ points on a circle. How many different triangles can be formed by choosing $3$ of these points as vertices?

    $56$

    $70$

    $28$

    $112$

11. A box contains $10$ different colored markers. In how many ways can you select $3$ markers?

    $30$

    $120$

    $720$

    $45$

12. A committee of $3$ men and $2$ women is to be formed from a group of $7$ men and $5$ women. How many different committees can be formed?

    $175$

    $35$

    $10$

    $350$

13. How many ways can you choose $3$ cards from a standard deck of $52$ cards such that all $3$ cards are spades?

    $286$

    $39$

    $1716$

    $220$

14. How many ways can you choose $2$ Kings and $1$ Queen from a standard deck of $52$ cards?

    $6$

    $24$

    $12$

    $8$

15. A team of $5$ is to be chosen from $6$ boys and $4$ girls. How many teams can be formed if there must be at least $4$ boys on the team?

    $30$

    $66$

    $60$

    $15$

16. A committee of $4$ people is to be formed from $5$ teachers and $6$ students. How many ways can the committee be formed if there are at most $2$ teachers?

    $225$

    $150$

    $250$

    $265$

17. Given that $\binom{n}{k} = \binom{n}{n-k}$, which of the following is equal to $\binom{15}{12}$?

    $\binom{15}{2}$

    $\binom{15}{3}$

    $\binom{15}{10}$

    $\binom{12}{3}$

18. How many unique sets of $3$ numbers can be chosen from the set $\lbrace 1, 2, 3, 4, 5, 6, 7 \rbrace$?

    $21$

    $210$

    $35$

    $49$

19. A coach needs to select $5$ players for a basketball team from a roster of $8$ guards and $6$ forwards. If the team must consist of $3$ guards and $2$ forwards, how many ways can the team be selected?

    $840$

    $720$

    $336$

    $560$

20. A box contains $9$ red balls and $7$ blue balls. In how many ways can you select $4$ balls such that at least $2$ of them are red?

    $1260$

    $630$

    $1350$

    $1470$

21. A committee of 5 is to be formed from 6 teachers and 8 students. In how many ways can this be done if the committee must include at least 1 teacher and at least 1 student?

    $2002$

    $1890$

    $1930$

    $1940$

22. A board of directors consists of 12 members. A subcommittee of 5 members is to be formed. If two specific members, Mr. Smith and Ms. Jones, refuse to serve together on the subcommittee, how many different subcommittees can be formed?

    $672$

    $572$

    $120$

    $792$

23. From a standard deck of 52 playing cards, a hand of 5 cards is dealt. How many distinct hands contain at most 2 kings?

    $2,596,704$

    $2,594,400$

    $2,598,960$

    $4,560$

24. A robot moves in a grid from point $(0,0)$ to point $(6,4)$. The robot can only move one step at a time, either to the right (R) or up (U). How many distinct paths are there?

    $15$

    $210$

    $3,628,800$

    $120$

25. A box contains 7 distinct red balls, 5 distinct blue balls, and 4 distinct green balls. In how many ways can a selection of 6 balls be made such that there are at least 2 red balls, at least 2 blue balls, and at least 1 green ball?

    $1260$

    $3500$

    $3080$

    $1400$

26. If $(n+1)C(n-1) = 28$, find the value of $n$.

    $8$

    $6$

    $7$

    $9$

27. How many distinct diagonals can be drawn in a regular polygon with 15 sides?

    $105$

    $15$

    $120$

    $90$

28. A committee of 6 is to be chosen from 9 men and 7 women. If the committee must have at least 3 men and at least 2 women, how many ways can the committee be formed?

    $5586$

    $2940$

    $8008$

    $1764$

29. A set of 5 distinct numbers is to be chosen from the set $\{1, 2, ..., 15\}$. How many such sets can be formed if at least two of the chosen numbers must be prime numbers?

    $756$

    $126$

    $3003$

    $2121$

30. A committee of 4 women and 3 men is to be chosen from 9 women and 7 men. In how many ways can this be done if two particular women refuse to serve together on the committee?

    $3675$

    $735$

    $4410$

    $3920$