The binomial coefficients count how many ways each kind of product appears.
WHAT HAPPENS IF…Why is the middle coefficient in (1+b)⁴ equal to 6?
Interactive exploration
Loading the mathematical model…
Connect the picture to the idea(a+b)n=k=0∑n(kn)an−kbk
Expanding n copies of (a+b) requires choosing a or b from each factor. Exactly k choices of b occur in C(n,k) ways. Grouping those equal products produces the theorem. This explorer fixes a=1 and varies b and the nonnegative integer n.
A detail that matters
The finite binomial theorem here assumes a nonnegative integer exponent.