Combinations

Lessons

1. Compare the difference between the following 2 scenarios:
   1. How many ways can a president, a vice president, and a treasurer be selected from a class of 20 students?
   2. How many different committees of 3 people can be selected from a class of 20 students?
2. A standard deck of 52 cards consists of:
   • 4 suits: diamonds, hearts, spades, and clubs
   • each suit has 13 cards
   • red cards: diamonds and hearts
   • black cards: spades and clubs
   • face cards: Jacks, Queens, and Kings
   How many different 5-card hands can be formed containing:
   1. any 5 cards (no restrictions)?
   2. all black cards?
   3. 1 black card and 4 red cards?
   4. all face cards?
   5. 3 Jacks and 2 Kings?
   6. 3 Jacks?
3. Problems with "at least", "at most"
   From a standard deck of 52 cards, how many different 5-card hands can be formed containing:
   1. exactly 2 Diamonds?
   2. at least 2 Diamonds?
   3. at most 2 Diamonds?
4. There are 20 members in a student council, 7 boys and 13 girls. How many ways can a committee of 5 people be selected if there must be:
   1. at least 3 boys?
   2. at most 1 girl?
   3. at least 1 girl?
   4. at least 2 boys and 2 girls?
5. Five points are marked on a circle. By connecting the points on the circle,
   1. How many triangles can be formed?
   2. How many line segments can be formed
   3. Form a pentagon inside the circle by connecting the 5 points. How many diagonals does the pentagon have?