Monday, January 21, 2013

The closed factorial form for the binomial coefficients

So far we have learned how to create Pascal's Triangle by using one row to generate the next.

1  1
1  2  1
1  3  3  1
1  4  6  4  1
1  5 10 10  5  1
1  6 15 20 15  6  1
1  7 21 35 35 21  7  1

What if we wanted to know how many different five card hands of poker there are. If we care about the suits of the cards, then "four aces and the king of spades" is different from "four aces and the king of clubs".  Counting that way the number of cards is "52 choose 5", the number of way to select five cards from a deck of fifty two.

Here is the closed form for the binomial coefficients. If this is the first time you have seen an exclamation point used in math, it means factorial.

n! = n(n-1)(n-2)(n-3)...(3)(2)(1), all the whole numbers from n to 1 multiplied together.  Let's do an example from part of the triangle we've already seen.

7 choose 2 is 7!/[2!5!]. notice that the 5×4×3×2×1 shows up in both the numerator and denominator, so it therefore can be cancelled out. We are left with 7×6/(2×1), which is 21.

 52 choose 5 will also have a lot of simple cancellations, and we end up with five consecutive numbers in the numerator divided by 5×4×3×2×1. When all the denominator is cancelled out, we are left with 2,598,960.

Tomorrow: Two problems from ancient India solved by numbers from the Triangle.

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