14.6 Combinations
Suppose you want to know how many combinations of looks you can do with the set of clothes that you have, what mathematical solution do you think you can use so you don’t have to mentally count all of the possible outcomes? How about, if you want to know how many kinds of arrangement you can do with the chairs in the classroom? In order to solve these problems you will need to know about permutations and combinations.

This chapter will have nine parts which will help us understand the basic concept of permutation and combination. In the first part of chapter, we will discuss about the fundamental counting principle. This principle states that if two separate events have m and m possible outcomes respectively, then the combined possible outcome for the combined events would be equivalent to m x n.

In the second part of this chapter will review how to write and solve a factorial notation. For example, if you are asked to solve four factorial or 4!, then you know that you need to solve for 1 x 2 x 3 x 4. Factorial notations are used throughout our discussion in this chapter like in proceeding parts of the chapter that focuses on path counting, and the difference between permutation and combination.

For the eighth this chapter, we will look into the application of all the concepts about permutations and combinations with that of the Pascal's triangle. If you want to know the answer for the question “what is a Pascal’s triangle?” you can check on it on various resources online.

Finally, in the last part of the chapter, we will look into the Binomial theorem. This theorem is used to find any power of a certain binomial without the need to multiply a long list of terms.


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!


? 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!}}
    • 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:
    • 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:
  • 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:
  • 5.
    Five points are marked on a circle. By connecting the points on the circle,
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