Question

Suppose that there are eight runners in a race. The winner receives a gold medal, the second-place finisher receives a silver medal, and the third-place finisher receives a bronze medal. How many different ways are there to award these medals, if all possible outcomes of the race can occur and there are no ties?

• A. \(56\)
• B. \(336\)
• C. \(126\)
• D. \(456\)

Hint:

Whenever positions, ranks, or medals are involved, order matters. Therefore use permutations: \[ {}^nP_r=\frac{n!}{(n-r)!}. \] If order does not matter, use combinations instead.

Answer 1

The Correct Option is B

Concept: This is a permutation problem because the medals are distinct. Gold, Silver and Bronze medals correspond to first, second and third positions respectively. Since changing the order changes the outcome, we must use permutations rather than combinations. The formula for permutations is \[ {}^nP_r=\frac{n!}{(n-r)!}. \] This gives the number of ways of arranging \(r\) objects selected from \(n\) distinct objects.

Step 1: Identify the total number of runners.
There are \[ 8 \] runners participating in the race. Out of these runners, only the first three positions are important.

Step 2: Assign the gold medal.
The gold medal can be awarded to any one of the eight runners. Therefore, \[ 8 \] choices are available.

Step 3: Assign the silver medal.
After awarding the gold medal, one runner has already been selected. Thus, \[ 7 \] runners remain eligible for the silver medal.

Step 4: Assign the bronze medal.
After awarding gold and silver medals, \[ 6 \] runners remain. Hence there are \[ 6 \] choices for the bronze medal.

Step 5: Apply the multiplication principle.
The total number of possible arrangements is \[ 8\times7\times6. \] \[ =336. \] Alternatively, \[ {}^8P_3 = \frac{8!}{5!} = 8\times7\times6 = 336. \]

Step 6: Write the final answer.
Hence the number of different ways to award the medals is \[ \boxed{336}. \] Therefore the correct option is \[ \boxed{(B)\;336}. \]