Chapter 9.6, Problem 74E

a.

To determine

Find the possible number of winning Powerball numbers.

Answer to Problem 74E

  $292,201,338$

Explanation of Solution

Calculation:

Consider the powerball game in which five white balls out of a drum of $69$ white balls (numbered $\text{1}–\text{69}$ ) and one red Powerball out of a drum of $\text{26}$ red balls (numbered $\text{1}–\text{26}$ ) are drawn. The jackpot is won by matching all five white balls in any order and the red Powerball.

The objective of this part is calculate the number of winning Powerball numbers.

The winning powerball numbers is achieved by selecting five white balls and one red Powerball.

Since, the order is not important, combinations are used to find the numbers of winning Powerball numbers. The number of ways of selecting $5$ white balls from $69$ white balls is ${}_{69}{C}_5$ . The number of ways of selecting $1$ red ball from $26$ red balls is ${}_{26}{C}_1$ .

So, the number of winning Powerball numbers is ${}_{69}{C}_5 {.}_{26}{C}_1$ .

Consider the following question,

  ${}_{69}{C}_5$

You know that the number of permutations of elements taken at a time is,

  ${}_n{C}_r=\frac{n!}{\left(n-r\right)!r!}\mathrm{..........}\left(1\right)$

  $C{}_5=\frac{69!}{\left(69-5\right)!5!}$

  $\begin{array}{l}=\frac{69!}{64!5!}\\ =\frac{69\times 68\times 67\times 66\times 65\times 64!}{64!\times 5\times 4\times 3\times 2\times 1}\\ =11,238,513\end{array}$

Also,

  $C{}_1=\frac{26!}{\left(26-1\right)!1!}$

  $\begin{array}{l}=\frac{26!}{25!1!}\\ =\frac{26\times 25!}{25!\times 1}\\ =26\end{array}$

Now, $\begin{array}{l}{}_{69}{C}_5 {.}_{26}{C}_1=\left(11,283,513\right)·\left(26\right)\\ =292,201,338\end{array}$

Hence, the possible number of winning Powerball numbers is $292,201,338$ .

b.

To determine

Find the possible number of winning Powerball number.

Answer to Problem 74E

  $1$

Explanation of Solution

Calculation:

The objective of this part is to find the possible number of winning Powerball numbers when you win the jackpot by matching all $5$ white balls in order and the red Powerball.

If the balls have to match the order, then there is only one possible sequence which will match the order. There is also one way of matching $1$ red Powerball.

So, the possible number of winning Powerball numbers to win jackpot is calculated as

  $1·1=1$

Hence, the number of possible winning Powerball numbers to win jackpot is $1$ .