Question: A sports team has 5 distinct positions. Out of 10 players, how many starting teams are possible if…

A sports team has 5 distinct positions. Out of 10 players, how many starting teams are possible if (A) The distinct positions are taken into consideration? (B) The distinct positions are not taken into consideration? (C) The distinct positions are not taken into consideration, but either Mike or Ken (but not both) must start? … (A) There are starting teams if the distinct positions are taken into consideration. (Type a whole number.)

Solution

To solve these problems, we’ll consider various scenarios for selecting a team from 10 players. (A) The distinct positions are taken into consideration: We are selecting 5 players for 5 distinct positions, so this is a permutation problem. 1. Calculate the permutations of selecting and assigning 5 players from 10. \[ P(10, 5) = \frac{10!}{(10-5)!} \] 2. Calculate further: \[ P(10, 5) = \frac{10 \times 9 \times 8 \times 7 \times 6}{1} = 30240 \] So, there are 30,240 starting teams if distinct positions are taken into consideration. (B) The distinct positions are not taken into consideration: This is a combination problem where we select 5 players out of 10 without considering positions. 1. Calculate the combination: \[ C(10, 5) = \frac{10!}{5!(10-5)!} \] 2. Calculate further: \[ C(10, 5) = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252 \] So, there are 252 starting teams if positions are not considered. (C) The distinct positions are not taken into consideration, but either Mike or Ken (but not both) must start: 1. First, count the combinations including Mike and excluding Ken: Select 4 other players from the remaining 8: \[ C(8, 4) = \frac{8!}{4!(8-4)!} = 70 \] 2. Next, count the combinations including Ken and excluding Mike: Select 4 other players from the remaining 8: \[ C(8, 4) = \frac{8!}{4!(8-4)!} = 70 \] 3. Add the two possibilities: Total teams = 70 + 70 = 140 So, there are 140 starting teams if either Mike or Ken (but not both) must start and positions are not considered.