2.3 ( ) www In this exercise, we prove that the binomial distribution (2.9) is normalized.

First use the definition (2.10) of the number of combinations of m identical objects chosen from a total of N to show that



N m



+



N m − 1



=



N + 1 m



. (2.262)

Use this result to prove by induction the following result (1 + x)N = N m=0 
N m 
xm (2.263)
which is known as the binomial theorem, and which is valid for all real values of x.
Finally, show that the binomial distribution is normalized, so that N m=0 
N m 
μm(1 − μ)N−m = 1 (2.264)
which can be done by first pulling out a factor (1 − μ)N out of the summation and then making use of the binomial theorem.