Pixar in a Box
- Start here!
- 1. Two headed robots
- Counting two-headed robots
- 2. Snake bots
- Building snake bots
- 3. Calculating factorials
- Calculating factorials
- 4. Casting problem
- Counting casts 1
- 5. Does order matter?
- Counting casts 2
- 6. Binomial coefficient
Get ready for a really powerful formula: the binomial coefficient! The binomial coefficient allows us to calculate the number of ways to select a small number of items from a larger group. The formula is represented as n choose k equals n! divided by k!(n-k)!. We can use it to solve problems like determining the number of possible casts from a group of actors.
Want to join the conversation?
- on0:50i dont get why our 6! / 3! became 6! / 3! * 3! ? where did the second 3! come from?(5 votes)
- That's because we need to divide by 3! to account for the 6 different permutations.
Remember it's: # combinations / # permutations.
# combinations = 6x5x4 = 6!/3!
# permutations = 3!
Let me know if you need more help(3 votes)
- Hi guys! Is it true that if k=0, the binomial coefficient equals 1? If so, then why? Thnx(2 votes)
- At1:15, if it's been a while since you took algebra, the reason n-(k-1) = n-k+1 is the distributive property. Not really a question, but it's something I had to look up and no one else asked it.(2 votes)
- would that mean 6! = 3!*3! ?(2 votes)
- I don't quite understand how this stuff works, any other videos i could watch first?
(i have seen allof the videos in this section)(2 votes)
- There is a tutorial on Khan Academy that might help: https://www.khanacademy.org/math/precalculus/prob_comb/combinatorics_precalc/v/factorial-and-counting-seat-arrangements(2 votes)
- at1:22can it also be n+k-1 ?(2 votes)
- Nice work. Thanks for stickin' with us. We're at the last step of the lesson. Earlier, I promised you a powerful counting formula. Let's work together to see if we can develop that formula. First notice that 6 x 5 x 4 looks a little like a factorial except that it's missing the 3 x 2 x 1. That means we can write 6 x 5 x 4 using factorials as 6! over 3!. Because 6! equals 6 x 5 x 4 x 3! So dividing by 3! just leaves 6 x 5 x 4. That means, we can rewrite our earlier example as 6! over 3! x 3!. To generalize this for other numbers of actors, let n be the number of actors we can pick from and let k be the size of the cast. On the first pick, we have n choices. Then, on the second pick, we have n-1 choices and so on. Notice that the number being subtracted is one less than the choice number. So, on the kth choice, you have n - (k-1) choices which is n - k +1. Multiplying the choices together gives n x n - 1 through n - k + 1 which can be written as n! over (n-k)!. Now, we have to divide by k! because there are k! ways to order the k choices. So, finally, we get to, wait for it. Drum roll, please! n! over k!(n-k)! possible casts of k actors chosen from a group of n actors total. This formula is so famous that it has a special name and a special symbol to write it. It's called a binomial coefficient and mathematicians write it as n choose k equals n! divided by k! (n-k)!. It's powerful because you can use it whenever you're selecting a small number of things from a larger number of choices. With this tool, we can easily compute, say, how many casts of 4 robots I can come up with when I have, let's say, 12 different robots to choose from. There are 12 choose 4, which, if you work it out, is exactly 495. Your final challenge, should you choose to accept it, is to answer some final questions with the binomial coefficient formula and there won't be any diagrams to help you this time. And, you'll be asked to count something other than robots, like, let's say, plants, or sandwiches, or outfits.