Questions: Answer the questions below. (a) An experiment involves 31 participants. From these, a group of 4 participants is to be tested under a special condition. How many groups of 4 participants are possible? (b) 75 athletes are running a race. A gold medal is to be given to the winner, a silver medal is to be given to the second-place finisher, and a bronze medal is to be given to the third-place finisher. Assume that there are no ties. In how many possible ways can the 3 medals be distributed?

Solution Steps

To determine how many groups of 4 participants can be formed from 31 participants, we use the combination formula:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

where \( n = 31 \) and \( r = 4 \).

\[ \binom{31}{4} = \frac{31!}{4!(31-4)!} = \frac{31 \times 30 \times 29 \times 28}{4 \times 3 \times 2 \times 1} \]

Calculating the numerator:

\[ 31 \times 30 \times 29 \times 28 = 756,840 \]

Calculating the denominator:

\[ 4 \times 3 \times 2 \times 1 = 24 \]

Now, divide the numerator by the denominator:

\[ \frac{756,840}{24} = 31,535 \]

To determine the number of ways to distribute the gold, silver, and bronze medals among 75 athletes, we use the permutation formula because the order matters:

\[ P(n, r) = \frac{n!}{(n-r)!} \]

where \( n = 75 \) and \( r = 3 \).

\[ P(75, 3) = \frac{75!}{(75-3)!} = 75 \times 74 \times 73 \]

Calculating the product:

\[ 75 \times 74 \times 73 = 405,150 \]

Final Answer