Question

A lottery game consists of drawing five distinct whole numbers from the numbers 1 through 61 in any order. Then one more number from the numbers 1 through 27 is selected as the final number (this number could be one of the original five). These numbers are drawn every Wednesday and Saturday. If you match all six numbers, you win the jackpot, which is worth at least $40 million. Use methods of this section to show that there are 160,626,969 possible plays.

(a) The first step is to select five distinct whole numbers between 1 and 61. Order is not important. Use the appropriate counting rule to determine the number of ways groups of five different numbers can be selected.

_______ ways?

(b) The next step is to choose the final number, which is any number between 1 and 27. The number need not be distinct from numbers chosen for the first five described in part (a). Use the appropriate counting rule to determine the number of possible distinct outcomes for the first five numbers, chosen as described in part (a) together with the final number.

__________ ways?

Answer 1

The true statements are:

• There are 5949147 ways, groups of first five different numbers can be selected.
• There are 160626969 possible distinct outcomes for the first five numbers, chosen as described in part (a) together with the final number

(a) The number of ways the first five numbers can be selected

The given parameters are:

$n = 61$ --- the total available number

$r = 5$ --- the numbers to select

Since the order of selection is not important, we make use of the combination formula as follows:

$^nC_r = \frac{n!}{(n -r)!r!}$

So, we have:

$^{61}C_5 = \frac{61!}{(61 -5)!5!}$

$^{61}C_5 = \frac{61!}{56!5!}$

Simplify

$^{61}C_5 = \frac{61 \times 60 \times 59 \times 58 \times 57}{5 \times 4 \times 3 \times 2 \times 1}$

$^{61}C_5 = \frac{713897640}{120}$

$^{61}C_5 = 5949147$

Hence, there are 5949147 ways, groups of first five different numbers can be selected.

(b) The total number of ways for all six numbers

The last number can be any number from 1 to 27.

So, the total number of ways is:

$Total = 27 \times 5949147$

$Total = 160626969$

Hence, there are 160626969 possible distinct outcomes

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