6. Binomial coefficient

nice work thanks for sticking 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 times 5 times 4 looks a little like a factorial except that it's missing the 3 times 2 times 1 that means we can write 6 times 5 times 4 using factorials as 6 factorial over 3 factorial because 6 factorial equals 6 times 5 times 4 times 3 factorial so dividing by 3 factorial just leaves 6 times 5 times 4 that means we can rewrite our earlier example as 6 factorial over 3 factorial times 3 factorial 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 minus 1 choices and so on notice that the number being subtracted is 1 less than the choice number so on the KA choice you have n minus K minus 1 choices which is n minus k plus 1 multiplying the choices together gives n times n minus 1 through n minus K plus 1 which can be written as n factorial over n minus K factorial now we have to divide by K factorial because there are K factorial ways to order the K choices so finally we get to wait for it drum roll please n factorial over K factorial times n minus K factorial 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 factorial divided by K factorial times n minus K factorial 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 four robots I can come up with when I have let's say twelve different robots to choose from there are twelve choose four 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