Expanding a bracket and choosing a team give the same numbers
Multiply out (1 + x) to the fourth power and the coefficients come out as 1, 4, 6, 4, 1. Ask how many ways there are to pick two items from four, and the answer is also 6. That coincidence is no accident: both questions are answered by binomial coefficients, read aloud as n choose k.
The link is easy to see once you picture the multiplication. Expanding (1 + x) to the power n means picking either the 1 or the x from each of n brackets. The number of ways to end up with exactly k copies of x is the number of ways to choose which k brackets supply them. So the coefficient of each power of x is a count of choices, and choosing 2 from the set 1, 2, 3, 4 gives the six pairs 12, 13, 14, 23, 24 and 34.
The numbers can be built by addition alone. Every entry equals the sum of two entries in the row above, which produces the triangular array known as Pascal's triangle. The reasoning is a neat piece of counting: fix one item, then split all the possible groups into those that include it, where you still need k minus 1 more from the remaining n minus 1, and those that leave it out, where you need all k from the rest.
There is also a direct formula, dividing n factorial by k factorial times n minus k factorial, or more efficiently multiplying k falling terms starting from n and dividing by k factorial. The same numbers answer many other puzzles, such as how many strings of n binary digits add up to k.
The ideas are old. Around 1150 the Indian mathematician Bhaskaracharya explained binomial coefficients in his book Lilavati, centuries before the stacked notation now used was introduced by Andreas von Ettingshausen in 1826. Calculators usually prefer a flat version such as nCr, with C standing for combinations, because it fits on a single line of display.
Source: Binomial coefficient
