Kata

The Binomial Form of a polynomial has many uses, just as the standard form does. For comparison, if p(x) is in Binomial Form and q(x) is in standard form, we might write

p(x) := a₀ * xC0 + a₁ * xC1 + a₂ * xC2 + ... + a_N * xCN

q(x) := b₀ + b₁ * x + b₂ * x² + ... + b_N * x^N

Both forms have tricks for evaluating them, but tricks should not be necessary. The most important thing to keep in mind is that aCb can be defined for non-integer values of a; in particular,

aCb := a * (a-1) * (a-2) * ... * (a-b+1) / b!   // for any value a and integer values b
    := a! / ((a-b)!b!)                          // for integer values a,b

The inputs to your function are an array which specifies a polynomial in Binomial Form, ordered by highest-degree-first, and also a number to evaluate the polynomial at. An example call might be

valueAt([1, 2, 7], 3);

value_at([1, 2, 7], 3)

value_at([1, 2, 7], 3)

and the return value would be 16, since 3C2 + 2 * 3C1 + 7 = 16. In more detail, this calculation looks like

1 * xC2 + 2 * xC1 + 7 * xC0 :: x = 3
3C2 + 2 * 3C1 + 7
3 * (3-1) / 2! + 2 * 3 / 1! + 7
3 + 6 + 7 = 16

More information can be found by reading about Binomial Coefficients or about Finite Differences.

Note that while a solution should be able to handle non-integer inputs and get a correct result, any solution should make use of rounding to two significant digits (as the official solution does) since high precision for non-integers is not the point here.