What is Combinatorics?
Combinatorics deals with mathematics surrounding counting. There are different "things" and methods to approaching this. There are also many applications of combinatorics in other areas of mathematics, such as graph theory, coding, probability, etc.
Factorials
We can count the number of orders in which something happens.
For example, if there are 3 students and 3 chairs, in how many different orders can the students sit on these chairs?
Solution: 6 differet possible orders
To simplify this, we can use factorials.
Try it yourself!
In how many different ways could 9 students sit on 9 chairs in a class?
Solution
Permutations
How do we approach the problem if there are not enough chairs?
We use for following formula
Important Permutation Reminders
Permutations are used if we care about the order
1! = 1
0! = 1
Try it yourself!
You forgot the code for your 4-digit lock. There are 10 digis (0-9) and each appears at most once. How many different ways can you try?
Solution
Combinations
Combinations are used when the order does not matter and you want to know how many ways there are to select a certain number of objects.
In a store, there are 5 different shirt colors: red, blue, green, yellow and black. You want to purchase 3 of them. How many ways are there to select 3 shirts out of the 5 that you like?
We use the following formula
Plug in the numbers...
Important Combination Reminders
Order does not matter
nCn = 1
nC0 = 1
nC1 = n
nCr = nC(n-r)
Try it yourself!
There are 10 students in your study group but you can only invite 5 to your party. How many different combinations of friends coul you invite?
(Note: use a pen and paper to help solve)
Solution
Test your knowledge
[qwiz repeat_incorrect=”false”; style=”border: 2px solid black; background: white;"] [q] Among a set of 5 black balls and 3 red balls, how many selections of 5 balls can be made such that at least 3 of them are black balls? [c] 15 [c*] 46 [c] 35 [f] Incorrect: Selecting at least 3 black balls from a set of 5 black balls in a total selection of 5 balls can be:
3 B and 2 R
4 B and 1 R and
5 B and 0 R balls.
5C3 * 3C2 + 5C4 * 3C1 + 5C5 * 3C0 = 46 ways [f]Correct!
5C3 * 3C2 + 5C4 * 3C1 + 5C5 * 3C0 = 46 ways [f]Incorrect: Selecting at least 3 black balls from a set of 5 black balls in a total selection of 5 balls can be:
3 B and 2 R
4 B and 1 R and
5 B and 0 R balls.
5C3 * 3C2 + 5C4 * 3C1 + 5C5 * 3C0 = 46 ways [q] In how many ways can a selection of 3 men and 2 women can be made from a group of 5 men and 5 women ? [c] 75 [c] 150 [c*] 100 [f] Incorrect - 5C3 * 5C2 = 100 [f] Incorrect - 5C3 * 5C2 = 100 [f] Corect! 5C3 * 5C2 = 100 [q] How many triangles can be formed by joining the verticies of an octagon? [c]112 [c*]56 [c]28 [f] Incorrect: Number of triangles can be formed by joining the verticies of a polygon of n sides - nC3 = 8C3 = 56 [f] Correct! Number of triangles can be formed by joining the verticies of a polygon of n sides - nC3 = 8C3 = 56 [f] Incorrect: Number of triangles can be formed by joining the verticies of a polygon of n sides - nC3 = 8C3 = 56 [q] Seven points are marked on a circle. How many quadrilaterals can be formed using 4 of the 7 points? [c] 15 [c] 20 [c*] 35 [f] Incorrect: 7C4 = 35 [f] Incorrect: 7C4 = 35 [f] Correct! 7C4 = 35 [q] In how many differeny ways can the letters of MISSISSIPPI be arranged? [c*] 34650 [c] 15120 [c] 48650 [f] Correct! (11!) / (4! 4! 2!) = 34650 [f] Incorrect: (11!) / (4! 4! 2!) = 34650 [f] Incorrect: (11!) / (4! 4! 2!) = 34650 [q] Using the digits 2, 3, 6, 8 and 9, how many 3-digit whole numbers can be formed if repititions are not permitted? [c*] 60 [c] 120 [c] 240 [f] Correct! - 5 numbers to choose from: 5 x 4 x 3 = 60 [f] Incorrect - 5 numbers to choose from: 5 x 4 x 3 = 60 [f] Incorrect - 5 numbers to choose from: 5 x 4 x 3 = 60 [q] There are three places P, Q and R such that 3 roads connects P and Q and 4 roads connects Q and R. In how many ways can one travel from P to R? [c] 8 [c] 10 [c*] 12 [f] Incorrect: 3 x 4 = 12 [f] Incorrect: 3 x 4 = 12 [f] Correct! 3 x 4 = 12 [q] How many permutations are there for all the letters in the word COMBINE? [c] 40320 [c] 10080 [c*] 5040 [f] Incorrect - 7 letters with no reptition: 7! = 5040 [f] Incorrect - 7 letters with no reptition: 7! = 5040 [f] Correct! - 7 letters with no reptition: 7! = 5040 [/qwiz]