Combinations

Combinations

How many choices do I have? This lesson can help you with answering this question! Combination is the process of selecting members from a set of items. Combination formula is your best friend in this lesson. Unlike permutations, order doesn't matter in combinations!

Lessons

? combination: nCr = number of selections of r items taken from a set of n distinct items (order does NOT matter!!)
= n!(nr)!r!\frac{{n!}}{{(n - r)!\;\;r!}}
  • 1.
    Compare the difference between the following 2 scenarios:
    a)
    How many ways can a president, a vice president, and a treasurer be selected from a class of 20 students?

    b)
    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:
    a)
    any 5 cards (no restrictions)?

    b)
    all black cards?

    c)
    1 black card and 4 red cards?

    d)
    all face cards?

    e)
    3 Jacks and 2 Kings?

    f)
    3 Jacks?

    g)
    flush (5 cards of the same suit)?


  • 3.
    ? problems with "at least", "at most"
    From a standard deck of 52 cards, how many different 5-card hands can be formed containing:
    a)
    exactly 2 Diamonds?

    b)
    at least 2 Diamonds?

    c)
    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:
    a)
    at least 3 boys?

    b)
    at most 1 girl?

    c)
    at least 1 girl?

    d)
    at least 2 boys and 2 girls?


  • 5.
    Five points are marked on a circle. By connecting the points on the circle,
    a)
    How many triangles can be formed?

    b)
    How many line segments can be formed

    c)
    Form a pentagon inside the circle by connecting the 5 points. How many diagonals does the pentagon have?