Question

1 point) 12 players for a softball team show up for a game: (a) how many ways are there to choose 10 players to take the field? (b) how many ways are there to assign people to the 10 different positions by selecting players from the 12 people who show up? (c) of the 12 people who show up, 2 are women. how many ways are there to choose 10 players to take the field if at least one of these players must be women?

Answer 1

To solve these problems, we utilize combinations for selecting players when order does not matter, and permutations for assigning players to specific positions. To ensure at least one woman is selected, we calculate the total possible teams and subtract the teams without women. The formulas for combinations C(n,k) and permutations P(n,k) are applied accordingly. a. 66 ways b. 66 ways c. 65 ways

Step-by-step explanation:

a. To choose 10 players out of 12, we can use the combination formula. The number of ways to choose 10 players out of 12 is given by C(12, 10) = 12! / (10! * (12-10)!) = 66 ways.

b. To assign players to the 10 different positions, we can use the permutation formula. The number of ways to select players for each position is given by 12P10 = 12! / (12-10)! = 12! / 2! = 66 ways.

c. To choose 10 players, including at least one woman, we need to consider two cases: (1) one woman and 9 men, and (2) two women and 8 men. For case (1), the number of ways to choose 1 out of 2 women and 9 out of 10 men is C(2, 1) * C(10, 9) = 2 * 10 = 20. For case (2), the number of ways to choose 2 out of 2 women and 8 out of 10 men is C(2, 2) * C(10, 8) = 1 * 45 = 45. Therefore, the total number of ways to choose 10 players with at least one woman is 20 + 45 = 65 ways.