A 9 passenger van shuttles athletes between venues at the canada summer games. If 22 athletes need to get to the track and field stadium, in how many ways can passengers be chosen for
A) the bus’s first trip?
B) the bus’s second trip?
Combinations
A) The bus's first trip can be chosen in $\binom{22}{9}$ ways.
B) The bus's second trip can be chosen in $\binom{13}{9}$ ways.
To determine the number of ways passengers can be chosen for each trip of the van, we will use the concept of combinations.
A combination is a set of items selected from a larger set, where the order of the items does not matter.
A) The bus's first trip:
Since there are 22 athletes who need to get to the track and field stadium, we want to select a combination of passengers from these 22 athletes for the first trip of the van.
Since it is mentioned that the van has a capacity for 9 passengers, we need to calculate the number of combinations of 9 athletes out of the 22 athletes.
The number of combinations can be calculated using the formula for combinations:
C(n, r) = n! / (r!(n-r)!),
where n is the total number of items and r is the number of items to be selected.
Using this formula, the number of combinations for the bus's first trip can be calculated as:
C(22, 9) = 22! / (9!(22-9)!) = 22! / (9! * 13!) ≈ 1,383,424.
Therefore, there are approximately 1,383,424 ways to choose the passengers for the bus's first trip.
B) The bus's second trip:
After the first trip, we would have 22 - 9 = 13 athletes remaining who still need to get to the track and field stadium.
Since the van has already made one trip, we now want to select a combination of passengers from these remaining 13 athletes for the second trip of the van, which also has a capacity for 9 passengers.
Using the same formula for combinations, the number of combinations for the bus's second trip can be calculated as:
C(13, 9) = 13! / (9!(13-9)!) = 13! / (9! * 4!) = 13 * 12 * 11 * 10 / 4 * 3 * 2 * 1 = 715.
Therefore, there are 715 ways to choose the passengers for the bus's second trip.
To summarize:
A) The bus's first trip: 1,383,424 ways to choose the passengers.
B) The bus's second trip: 715 ways to choose the passengers.