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I am taking a statistics course and found this question in my textbook. I am trying to incorporate the binomial theorem into the proof, but am hitting a wall. Any suggestions would be greatly appreciated.

Update: Is this answer sufficient or would one argue that I need to include more steps?

By the definition of the binomial theorem, (n choose k)(p)^(k)(1-p)^(n-k)= (x+y)^n If you set x=p, and y=(1-p), then x+y = 1. The limit of 1^(n) as n approaches infinity is 1, therefore the sum of (n choose k)(p)^(k)(1-p)^(n-k)=1.

Greta
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Do you know the binomial formula

$$\sum_{k=0}^n{n\choose k}a^kb^{n-k}=(a+b)^n\quad?$$ Apply it with $a=p$ and $b=1-p$.