Derive and use binomial results

There are C(n,x) sequences with x successes, and each has probability p raised to the power of x ^x(1-p) raised to the power of n-x ^n-x; multiply them. The binomial theorem makes the masses sum to [p+(1-p)] raised to the power of n ⁿ=1. Mean and variance follow by adding the moments of independent Bernoulli indicators. At most r means X≤r; at least r means X≥r. Use a complement where it reduces the number of terms.