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 your 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 your answer.)

Permutation Formula: We are looking to find the number of different ways to award three distinct prizes to $22$ people. This is a permutation problem because the order in which the prizes are awarded matters (first, second, and third are distinct). $\newline$ To calculate this, we use the permutation formula $P(n, k) = \frac{n!}{(n-k)!}$ , where $n$ is the total number of people, and $k$ is the number of prizes. $\newline$ Here, $n = 22$ and $k = 3$ .
Calculate Factorial: First, we calculate the factorial of $n$ , which is $22!$ ( $22$ factorial). However, we do not need to calculate the entire factorial for $22$ because it will be partially canceled out by the $(22-3)!$ in the denominator.
Calculate $(n-k)!$ : Next, we calculate $(n-k)!$ , which is $(22-3)!$ or $19!$ . Again, we do not need to calculate the entire factorial for $19$ because it will cancel out with part of the $22!$ in the numerator.
Calculate Permutation: Now, we calculate the permutation $P(22, 3) = \frac{22!}{(22-3)!}$ . This simplifies to $P(22, 3) = \frac{22 \times 21 \times 20}{19 \times 18 \times 17 \times \ldots \times 1}$ , where all the terms from $19$ to $1$ cancel out.
Perform Multiplication: After canceling out the common terms, we are left with $P(22, 3) = 22 \times 21 \times 20$ . Now we perform the multiplication: $22 \times 21 \times 20 = 462 \times 20 = 9240$ .
Final Result: Therefore, there are $9240$ different ways in which the prizes can be awarded.