## Precalculus (6th Edition) Blitzer

a) The probability of winning the prize with one lottery ticket is $\frac{1}{15,504}$. b) The probability of winning the prize with $100$ different lottery tickets is $\frac{25}{3876}$
(a) We know that the number of ways in which $5$ different numbers can be chosen from $1$ to $20$ is given below: \begin{align} & _{20}{{C}_{5}}=\frac{20!}{5!\left( 20-5 \right)!} \\ & =\frac{20!}{5!15!} \\ & =\frac{20\times 19\times 18\times 17\times 16\times 15!}{5!15!} \\ & =15504 \end{align} Therefore, $n\left( S \right)=15504$ Now, there is only one way to win the lottery. Therefore, $n\left( E \right)=1$ Then, \begin{align} & P\left( E \right)=\frac{n\left( E \right)}{n\left( S \right)} \\ & =\frac{1}{15,504} \end{align} Thus, the probability of winning the prize with one lottery ticket is $\frac{1}{15,504}$. (b) We know that the number of ways in which $5$ different numbers can be chosen from $1$ to $20$ is given below: \begin{align} & _{20}{{C}_{5}}=\frac{20!}{5!\left( 20-5 \right)!} \\ & =\frac{20!}{5!15!} \\ & =\frac{20\times 19\times 18\times 17\times 16\times 15!}{5!15!} \\ & =15504 \end{align} Therefore, $n\left( S \right)=15504$ There are $100$ different tickets so, the number of ways of choosing $1$ from $100$ different tickets is ${}^{100}{{C}_{1}}=100$. Therefore, the probability of winning the prize with one lottery ticket is \begin{align} & P\left( E \right)=\frac{n\left( E \right)}{n\left( S \right)} \\ & =\frac{100}{15504} \end{align} Thus, the probability of winning the prize with one lottery ticket is $\frac{25}{3876}$.