Solve by the method of your choice. $\newline$ Twenty-two people purchase raffle tickets. Three winning tickets are selected at random. If first prize is $\$ 1000$ , second prize is $\$ 500$ , and third prize is $\$ 100$ , in how many different ways can the prizes be awarded? $\newline$ There are $\square$ different ways in which the prizes can be awarded. $\newline$ (Simplify you answer.)

Full solution

Q. Solve by the method of your choice.

\newline

Twenty-two people purchase raffle tickets. Three winning tickets are selected at random. If first prize is

\$ 1000

, second prize is

\$ 500

, and third prize is

\$ 100

, in how many different ways can the prizes be awarded?

\newline

There are

\square

different ways in which the prizes can be awarded.

\newline

(Simplify you answer.)

Understand the problem: Understand the problem. $\newline$ We need to find the number of different ways to award three distinct prizes among $22$ people. This is a permutation problem because the order in which the prizes are awarded matters.
Calculate first prize: Calculate the number of ways to award the first prize. $\newline$ There are $22$ people and only one can win the first prize, so there are $22$ possible ways to award the first prize.
Calculate second prize: Calculate the number of ways to award the second prize. $\newline$ After the first prize has been awarded, there are $21$ people left. Therefore, there are $21$ possible ways to award the second prize.
Calculate third prize: Calculate the number of ways to award the third prize. $\newline$ After the first and second prizes have been awarded, there are $20$ people left. Hence, there are $20$ possible ways to award the third prize.
Calculate total ways: Calculate the total number of different ways to award all three prizes. $\newline$ To find the total number of different ways to award all three prizes, we multiply the number of ways to award each prize together: $22$ ways for the first prize, $21$ ways for the second prize, and $20$ ways for the third prize. $\newline$ Total number of ways = $22 \times 21 \times 20$
Perform multiplication: Perform the multiplication to find the final answer. $\newline$ Total number of ways = $22 \times 21 \times 20 = 9240$