Binomial Theorem

In Probability,

$$(p+q)^n = \sum\limits_{k=0}^n {n \choose k} p^k q^{n-k}$$

Proof

Base case

Let $n=1$. Then

$$\sum\limits_{k=0}^1 {1 \choose k }p^k q^{n-k} = {1 \choose 0} p^0 q^1 + {1 \choose 1} p^1 q^0$$

Induction Hypothesis

Assume that $(p+q)^m = \sum\limits_{k=0}^m {1 \choose m }p^k q^{m-k}$. Let $n = m+1$. Then

$$(p+q)^{m+1} = (p+q) (p+q)^m = (p+q) \sum\limits_{k=0}^m {1 \choose m }p^k q^{m-k}$$

$$p \sum\limits_{k=0}^m {m \choose k}p^k q^{m-k} + q \sum\limits_{k=0}^m {m \choose k}p^k q^{m-k}$$

$$\sum\limits_{k=0}^m {m \choose k}p^{k+1} q^{m-k} + \sum\limits_{k=0}^m {m \choose k}p^k q^{m-k+1}$$

Let $j=k+1$ and $k=j-1$ for the first and $j=k$ for the second sum. Then,

$$\sum\limits_{j=1}^{j=m+1} {m \choose {j-1}}p^{j} q^{m + 1 - j} + \sum\limits_{j=0}^{j=m} {m \choose j}p^j q^{m-j+1}$$

$${m \choose m} p ^{m+1} q^0 + \sum\limits_{j=1}^{j=m} {m \choose {j-1}} p^j q^{m+1-j} + \sum\limits_{j=1}^{j-m} {m \choose j} p^j q^{m+1-j} + {m \choose 0} p^0 q^{m+1}$$

This equals

$${m+1 \choose m+1} p ^{m+1} q^0 + \sum\limits_{j=1}^{j=m} ({m \choose {j-1}} + {m \choose j}) + \sum\limits_{j=1}^{j-m} {m \choose j} p^j q^{m+1-j} + {m+1 \choose 0} p^0 q^{m+1}$$

$${m+1 \choose m+1} p ^{m+1} q^0 + \sum\limits_{j=1}^{m} {m+1 \choose j} p^j q^{m+1-j} + {m+1 \choose 0} p^0 q^{m+1}$$

$$= \sum\limits_{j=0}^{m+1} {m+1 \choose j} p^j q^{m+1-j} ~~~~~~~~~~ \blacksquare$$