Question
Question 10 (1 point) Sam is the captain of an academic team in the class. A teacher will choose 5 of the 30 students in the class to be on an academic team. How many ways can 5 students be chosen as a team from this class given that Sam must be one of those students? Your Answer:
Ask by Ward Bartlett. in the United States
Feb 17,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
There are 23,751 ways to choose the team with Sam included.
Solution
Since Sam must be on the team, we are only choosing 4 additional students from the remaining 29. The number of ways to do this is given by "29 choose 4", which is calculated as:
C(29, 4) = (29 × 28 × 27 × 26) / (4 × 3 × 2 × 1)
Evaluating this gives the final answer.
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Simplify this solution Beyond the Answer
To find the number of ways to choose the academic team with Sam already included, we need to select 4 additional students from the remaining 29 classmates. The number of combinations can be found using the combination formula: \[ C(n, r) = \frac{n!}{r!(n - r)!} \] In this case, we have \( n = 29 \) and \( r = 4 \): \[ C(29, 4) = \frac{29!}{4! \cdot (29 - 4)!} = \frac{29 \times 28 \times 27 \times 26}{4 \times 3 \times 2 \times 1} = 23751. \] So, there are 23,751 ways to choose 5 students with Sam as one of them.
